(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Littoral sand transport on beaches"

Copyright 19 7 9 
Todd L. Walton Jr. 



Lif^oiAL Bmj3 immsmm ow benches 



TQDD L. WALTON JR. 



A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF 
THE UNIVERSITY OF FLORIDA 
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE 

DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 



ACKNOWLEDGEMENTS 

With utmost sincerity, the author wishes to thank Dr. 
B.A. Christensen and Dr. T. Y. Chiu for their advice, direc- 
tion, and considerable assistance which they have given to 
the author's Ph.D. program. Without their untiring help and 
encouragement this thesis would not have been possible. 

Thanks are also due to Dr. D. P. Spangler, Dr. F. W. 
Morris, and Dr. J. G. Melville, for serving as the author's 
supervisory committee. 

The author also wishes to express his deep appreciation 
to his former mentor, Dr. R. G. Dean whose efforts in helping 
the author to understand the basics of science have never 
ceased, and to his good friend the late Professor J. A. 
Purpura whose encouragement and personal understanding of 
shore processes were of tremendous benefit to the author. 

Appreciation is also extended 'to the Florida Sea Grant 
Program whose grant C0M--3-158-43 ) made many of the early 
results of this thesis possible, and the Department of 
Natural Resources, Bureau of Beaches and Shores, State of 
Florida, whose support and encouragement have benefitted the 
author . 

Finally, the author wishes to thank his wife Cindy 

whose patience and unselfishness throughout the pursuit of 

this dissertation has made life more wonderful. 

iii 



TABLE OF CONTENTS 



Page 

ACKNOWLEDGMENTS iii 

LIST OF TABLES vii 

LIST OF FIGURES ix 

LIST OF SYMBOLS xvi 

ABSTRACT xxiv 

CHAPTER 

I. INTRODUCTION 1 

1-1 Introductory Note 1 

1.2 Statement of Erosion Problem 3 

1.3 Objectives 17 

II. REVIEW OF SURF ZONE MODELS 19 

2.1 Review of Existing Longshore Sand 
Transport Models 19 

2 . 2 Review of Existing Longshore 

Current Models 2 7 

2.3 Review of Friction Factor 39 

2.3.1 Field Data i+0 

2.3.2 Laboratory Data i44 

III. REVIEW OF EXISTING DATA ON LONGSHORE 

SAND TRANSPORT 51 

3.1 Introductory Remarks 51 

3.2 Laboratory Studies 51 

3.2.1 Krumbein Laboratory Study 53 

3.2.2 Seville Laboratory Study 5 6 

3.3.3 Shay and Johnson Laboratory Study . 56 
3 . 3 . M- Sauvage and Vincent Laboratory 

Study 5 8 

3.3.5 Savage and Fairchild Laboratory 

Study 5 8 

3.3.6 Price, Tomlinson, and Willis 
Laboratory Study 6 0 

3.2.7 Barcelo Laboratory Study 61 

3.2.8 Bijker Laboratory Study 61 



IV 



Page 



3.3 Field Studies 62 

3.3.1 Watt Field Study 5 5 

3.3.2 Caldwell Field Study 69 

3.3.3 CERC Field Study 70 

3.3.4 Moore and Cole Field Study .... 70 

3.3.5 Komar Field Study 71 

3.3.6 Thornton Study 7 2 

IV. SAND TRANSPORT MODEL DEVELOPMENT 7 3 

4-.1 Introduction 7 3 

M- . 2 Review of Mass, Momentum, and Energy 

Conservation in Periodic Wave Flow ... 76 

4.3 Review of Wave Field Equations 87 

4.14 Proposed Sand Transport Model 105 

4.4.1 Longshore Current Estimation . , .10 5 

4.4.2 Integrated Sand Transport 
Estimation 120 

4.4.3 Sand Transport Distribution 

Across Surf Zone 131 

4.4.4 Estimation of Efficiency 
Factors and Dynamic Friction 
Coefficient 13 8 

4.4.5 Model Comparison with Laboratory 

and Field Results 155 

V. APPLICATION OF SAND TRANSPORT EQUATION 
TO CALCULATION OF LITTORAL DRIFT 

USING SHIP WAVE DATA 16 2 

5.1 Application of Model 16 2 

5.1.1 Data Source' 16 4 

5.1.2 Analysis of Wave Data to 

Compute Sand Transport 16 5 

5.2 Results of Sand Transport 

Computations 17 2 

5.2.1 Use of a Littoral Drift Rose . . . 175 

5.2.2 Possible Sources of Data 

Error or Bias 178 

5.2.3 Other Possible Errors or 

Bias in Analysis 180 

5.2.4 Other Potential Sources 

of Error 182 

5 . 3 Comparison of Calculated Littoral 
Drift Rates with Previously 

Estimated Values 185 

5.4 Comparison of Estimated and 

Observed Wave Climates 19 8 

VI. ANALYTICAL MODELS FOR SHORELINE CHANGE . . . 20 8 

6.1 Introductory Rem^arks 208 



V 



Page 



6 . 2 Heuristic "Equilibrium" 

Shoreline Model Development 209 

6 . 3 , Analytical Treatment of Shoreline 

Change Model 229 

VII. SAND TRANSPORT AND STORAGE AT INLETS .... 248 

VIII. SUMMARY AND CONCLUSIONS 259 

REFERENCES 262 

APPENDIX 

A. DERIVATION OF LONGSHORE ENERGY FLUX 

PER UNIT LENGTH SHORELINE 275 

B. LONGSHORE SAND TRANSPORT MODEL DATA 

SUSPENDED SAND CONCENTRATION DATA 2 80 

C. ANALYSIS OF SSMO WAVE HEIGHT, PERIOD, 

AND DIRECTION RANGES 289 

BIOGRAPHICAL SKETCH . ' ■ . 321 



vi 



LIST OF TABLES 



Table Page 

1.1 Volumes of Material Present in Outer 

Inlet Shoals of Florida Inlets . 11 

1.2 Beach Nourishment Projects 12 

2.1 Littoral - Sand Transport, .Models 20 

2.2 Littoral Sand Transport Models of 

Equation Type 2.1 23 

2.3 Longshore Current Formulas 29 

2.4- Results of Field Measurements of Fric- 
tion Factor i43 

2.5 Results of Laboratory Measurements of 

Friction Factor i+8 

3.1 Testing Variables of Laboratory Data 54 

3.2 Summary of Wave Basin Dimensions 55 

3- 3 Field Study Tests for Sand Transport 

Determination 56 

4- .1 Observed and Theoretical Values of 



4.2 Integrals for Sediment Transport 137 

5.1 Comparison of Annual Net Littoral Drift 

Rates as Estimated by the U.S. Army Corps 
of Engineers and as Calculated in the 

Present Study 183 

5.2 Comparison of Annual Average Total Lit- 

toral Drift Rates as Estimated by the 
U.S. Army Corps of Engineers and as 

Calculated in the Present Study 187 



vi.i 



LIST OF TABLES 
(Continued) 

Table Page 

5.3 Recording Periods of Shore Based CERC 

Wave Gages Used in Comparison of Actual 

to Predicted Shore Wave Climate 200 

B.l Shay and Johnson Data 280 

B.2 Komar Data 281 

B.3 Fairchild Data 282 

B.4 Watts Data 283 

B.5 Kana Data 2B^■ 

B. 6 Fairchild Data 285 

C. l Representative Values of Wave Height Used 

in Computation of Longshore Energy Flux 291 

C.2 Representative Values of Wave Period Used 

in Computation of Longshore Energy Flux 292 



::v ill 



LIST OF FIGURES 

-^^g^^ Page 

1.1 Erosion Situation in Florida 2 

1.2 Relative Rise in Sea Level in Florida t| 

1.3 Bruun's Concept of Beach Profile 

Response to Sea Level Rise 5 

1.4 Sediments Deposited off Central Padre 
Island During and After Hurricane Carla 

on a Mud Bottom 7 

1.5 General Beach-Dune Profile Before and 
After Hurricane Eloise Between Panama 

City and Destin, Florida 9 

1.5 Deposition of Sand in the Interior of 

St. Lucie Inlet 14. 

1.7 Relationships Between Outer Bar Stor- 
age and Cross Sectional Area of Inlets 15 

3.1 Distribution of Longshore Velocity and 

Sediment Transport Ac:toss the Surf Zone , 61+ 

14.1 Shear Stresses Acting on the Faces of 

a Water Column 84 

4.2 Schematic of the Sur;' Zone 92 

4.3 Schematic Plan and Section of the Near- 
shore Region 108 

4.4 Shear Forces Due to Oscillatory Wave 

Velocity and Longshore Current 115 

4.5 Relation of Normal Force to Moving Force ...... 123 

4.5 Plot of Sand Distr'bution Across the 

Surf Zone 135 

4.7 Sediment Suspension Due to Cb) Long 

Periods, High Fall Velocities, (c) Short 

Periods, Low Fall Velocities li+O 



LIST OF FIGURES 
( Continued ) 



Figure Page 

4.8 Vertical Forces on a Grain Particle 

About to be Lifted from the Bed 144 

4.9 Relationship Between Sediment Mixing 

Coefficient and Mixing Parameter — g — ~ 150 

4.10 Relationship Between Sediment Concentra- 
tion and Lift to Weight Ratio Parameter 151 

4.11 Postulated Values of Efficiency 

Factor e, 154 

a 

4.12 Variation of Dynamic Friction Angle 

with Solids Reynold's Number 155 

4.13 Sand Transport Model Relationship 158 

4.14 Best Fit Sand Transport Model for 

Fairchild Data 159 

4.15 Best Fit Sand Transport Model for 

Field Data 160 

5.1 Definition of Azimuth Angle Normal to 
Shore 9 , and Azimuth Angle of Wave 

Propagation 9 168 

5.2 Relationship Between Direction of Wave 
Propagation and Direction of Longshore 

Sand Transport 171 

5.3 Location of SSMO Data Squares Adjacent 

to the Florida Peninsula 173 

5.4 Azimuth of Normal to Shoreline at Ponte 

Vedra Beach, Florida 175 

5.5 Determination of Total Positive and Total 

Negative Littoral Sand Transport 177 

5.6 Ideal Case of an Unstable Null Point 191 

5.7 Ideal Case of a Stable Null Point 193 



LIST OF FIGURES 
(Continued) 



FigTjre . Page 

5.8 Stable Type Littoral Drift Rose Due to 

Unimodal Wave Climate 195 

5.9 Unstable Type Littoral Drift Rose Due to 

Bimodal Type Wave Climate 196 

5.10 Instability Formed Capes in Santa Rosa 

Sound 197 

5.11 Comparison of Computed and Observed Wave 

Heights at Daytona Beach, Florida 201 

5.12 Comparison of Computed and Observed Wave 

Periods at Dajrtona Beach, Florida 202 

5.13 Comparison of Computed and Observed Wave 

Heights at Lake Worth-Palm Beach, Florida 203 

5.14 Comparison of Computed and Observed Wave 

Periods at Lake Worth-Palm Beach, Florida 2Q^■ 

5.15 Comparison of Computed and Observed Wave 

Heights at Naples, Florida 205 

5.16 Comparison of Computed and Observed Wave 

Periods at Naples, Florida 205 

6.1 Logspiral Curve Fit to Shoreline at 

Riomar, Florida 211« 

6.2 Logspiral Curve Fit to Shoreline at 

Lacosta Island 212 

6.3 Logspiral Curve Fit to Panhandle Coast . 213 

6.1+ Orientation of a Shoreline Segment Having 

Equal Offshore Wave Energy 217 

6.5 Wave Energy Rose for Ocean Data Square 

#12 off East Coast of Florida 219 

6.6 Energy Diagram of Coastline with Equilib- 
rium Coast Shape 221! 



LIST OF FIGURES 
(Continued ) 



Fig^e Page 

6.7 Solution Curves to Equilibrium Shoreline 

Equation for the East Coast of Florida 222 

5.8 Equilibrium Shoreline Shape for Riomar, 

Florida 22l| 

6.9 Solution Curves to Equilibrium Shoreline 

for Gulf Coast of Florida 226 

5.10 Equilibrium Shoreline Shape for Lacosta 

Island, Florida 22? 

5.11 Equilibrium Shoreline Shape for Panhandle 

Coast of Florida 228 

6.12 Equilibrium Shoreline Orientation for 
Shoreline Protected by Offshore Break- 
waters on the East Coast of Florida 230 

6.13 Definition of Sign Convention for Shore- 
line Modeling 233 

6.14 Solution Curve for Case of Sand Buildup 

at a Structure 239 

6.15 Solution Curve for Sand Buildup at Struc- 
ture (After Time Sand has Reached End of 

Structure ) 243} 

6.16 Beach Nourishment Plan Shapes for Beach 

Fill 242 

5.17 Beach Nourishment Plan Slopes for Triangu- 
lar Initial Distribution 2m 

6.18 Beach Noiarishment Plan Shapes for Gap in 

Beach Fill , 24S 

6.19 Solution Graph for One End of Semi-infinite 

Beach Nourishment Project 245 

7.1 Tidal Pr ism— Outer Bar Storage Relationship 

for Highly Exposed Coasts 25S 

xii. 



Copyright 19 7 9 
by 

Todd L. Walton Jr, 



LIST OF FIGURES 
(Continued) 



Figiire Page 

7.2 Tidal Prism-Outer Bar Storage Relationship 

for Inlets on Sandy Coasts 25U 

7.3 Tidal Prism-Outer Bar Storage Relationship 

for Moderately Exposed Coasts 253' 

7.1+ Tidal Prism-Outer Bar Storage Relationship 

for Mildly Exposed Coasts 255 

A.l Definitions for Conservairion of Energy 

Flux for Shoaling Wave 276 

C.l Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Fernandina 
Beach to St, John's River, Florida 295 

C.2 Variation of Average Annual Total Littoral 

Drift with Beach Orientation — St. John's 

River to St. Augustine Inlet, Florida 296 

C.3 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — St. Augus- 
tine Inlet to Ponce de Leon Inlet, Florida 297 

C.^■ Variation of Average Annual Total Littoral 

Drift with Beach Orientation — Ponce de 

Leon Inlet to Cape Kennedy, Florida 298 

C.5 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Cape Kennedy 
to Sebastian Inlet, Florida 299 

C.6 Variation of Average Annual Total Littoral 

Drift with Beach Orientation — Sebastian 

Inlet to Fort Pierce Inlet, Florida 300 

C.7 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Fort Pierce 
Inlet to St, Lucie Inlet, Florida 301 

C.8 Variation of Average Annual Total Littoral 

Drift with Beach Orientation — St. Lucie 

Inlet to Jupiter Inlet, Florida 302 



xxii 



LIST OF FIGURES 
(Continued) 



Figure Pag^ 

C.9 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Jupiter Inlet 
to Lake Worth Inlet, Florida 303 

C.IO Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Lake Worth 
Inlet to Hillsboro Inlet, Florida 304 

C.ll Variation of Average Annual Total Littoral 
Drift with Beach Orientation — ^JiilJ_sboro 

Inlet to Cape Florida, Florida 305 

C.12 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Perdido Pass 
to Pensacola Bay Entrance, Florida 306 

C.13 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Pensacola 
Bay Entrance to Choctawhatchee Bay 

Entrance, Florida 307 

C.14 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Chocta- 
whatchee Bay Entrance to St. Andrew Bay 

Entrance, Florida 308 

C.15 Variation of Average Annual Total Littoral 

Drift with Beach Orientation — St. Andrew 
Bay Entrance to St. Joseph Bay Entrance, 

Florida 309 

C.16 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — St. Joseph 
Bay Entrance to Cape San Bias, Florida 310 

C.17 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Cape San 

Bias to Cape St. George, Florida 311 

C. 18 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Anclote 

Keys to Clearwater Pass, Florida 312 

C.19 Variation of Average Annual Total Littoral 

Drive with Beach Orientation — Clearwater 

Pass to Tampa Bay Entrance, Florida 313 

xiv 



LIST OF FIGURES 
(Continued) 



^ig^e Page 

C.20 Variation of Average Annual Total Littoral 

Drift with Beach Orientation — Tampa Bay 

Entrance to Big Sarasota Pass, Florida 314 

C,21 Variation of Average Annual Total Littoral 
Drift with Beach Orientation- -Big Sara- 
sota Pass to Venice Inlet, Florida 315 

C.22 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Venice Inlet 
to Boca Grande Inlet, Florida 315 

C.23(a) Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Boca Grande 
Inlet to San Carlos Bay, Florida 317 

C. 23(b) Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Boca Grand 
Inlet to San Carlos Bay, Florida 318 

C.21+ Variation of Average Annual Total Littoral 
Drift with Beach Orientation — San Carlos 

Bay to Wiggins Pass, Florida 319 

C.25 Variation of Average Annual Total Littoral 
Drift with Beach Orientation — Wiggins Pass 
to Cape Romano, Florida 320 

C. 26 Modification of Wave Data for Waves Paral- 
lel to Coastline 293 



LIST OF SYMBOLS 

A = mixing parameter constant 

= inlet cross sectional area at throat of tidal inlet 
a = wave amplitude 

B = proportionality factor associated with bed load 
transport 

Bg = proportionality factor associated with bed load 
transport inside the surf zone 

B^ = constants of longshore velocity distribution 

^i' ^i ~ functions of bed load portion of sand transport 
equation. 

c = wave celerity 

c^ = wave celerity at wave breaking 

Cg = speed of wave energy propagation 

c^ = wave celerity in deep water 

C = concentration in dry weight of sediment per unit 
weight fluid 

= drag coefficient 

= friction coefficient for flat plates 
C-j^ = life coefficient 

= Chezy friction coefficient 
d = mean sand grain size 
dgp = median sand grain size 

D = n + H, total depth of water 

XV i 



= total depth of water at location of breaking waves 

= depth of water at limits of longshore sand movement 

e^ = efficiency coefficient associated with bed load 
transport 

e^ = efficiency coefficient associated with suspended 
load transport 

E = energy density 
E* = quantity proportional to energy flux density 
E^ = energy density in deep water 

f = friction factor 

f = frequency of occurrence 

f^ = friction factor defined by energy dissipation for 
waves 

f^ = friction factor defined for oscillatory flow 

= functions of sand transport equation 

F = energy density flux 

F . = horizontal force per unit area due to slope of 
'^'^ free water surface 

g = acceleration due to gravity 

G = gravity force on a spherical sediment particle 

h = depth below still water 

hj^ = depth below still water level at wave breaking 

h^^ = height of beach berm above still water level 

h^ = depth below still water level in deep water 

h^ = total depth of active beach profile (in longshore 
sand transport) 

H = wave height 

H, = wave height at wave breaking 



xvii 



■ wave height in deep water 

■ significant wave height 

index corresponding . to horizontal coordinate in 
the x-di^^^ection 

inmersed weight sediment transport rate per unit 
width 

immersed weight sediment transport rate per unit 
width (bed load) 

immersed weight sediment transport rate per unit 
width (suspended load) 

total immersed weight sediment transport rate for 
surf zone 

total immersed weight sediment transport rate for 
surf zone (bed load) 

total immersed weight sediment transport rate for 
surf zone (suspended load) 

index corresponding to horizontal coordinate in 
the y-direction 

wave number 

ratio between, set-up slope and beach slope 
constants 

friction coefficient 
refraction coefficient 
shoaling coefficient 
rip current spacing 
wave length 

lift force on a spherical sediment particle 
wave length in deep water 
mass of bed load sediments 



X viii 



m = mass of suspended sediments 

M = summation constant 

= + M^, total mass transport per unit width or 
total mean momentum per unit area 

= mass transport per unit width associated with 
fluctuating motion 

n = transmission coefficient 

N = dimensionless lateral mixing coefficient 

N* = normal directed, component of energy density at 
shoreline 

p = pressure 

p = dimensionless lateral mixing coefficient 

p^ = porosity 

= power/unit area available for bed sediment 
transport 

= "longshore" energy flux 

Pg = power/unit area available for suspended sediment 
transport 

P* = parallel directed component of energy density at 
shoreline 

Qi^ = total volumetric longshore sand transport rate 

r = roughness parameter 
r^ = ripple parameter 

= total average resistance force 

R = ripple height 

R = equivalent of Reynolds Number in solids mixture 
s 

s = fraction of incoming energy flux available to the 
longshore current 



xix 



proportionality factor associated with suspended 
load transport 

specific gravity of sediment 

proportionality factor associated with suspended 
load transport inside the surf zone 

excess momentum flux tensor (radiation stress 
tensor) 

function of suspended load portion of sand trans- 
port equation 

time 

wave period 

gradient of "radiation" stress due to periodic 
wave motion 

water particle velocity maximum due to wave motion 
at the bed 

total water particle velocity 

fluctuating water particle velocity component 

water particle velocity due to wave motion 

mean velocity, component normal to the beach 

mean velocity of bed load transport in the long- 
shore direction 

mean velocity of suspended load transport in the 
longshore direction 

+ , total mean transport velocity 

mean transport velocity associated with mean 
motion 

mean transport velocity associated with fluctu- 
ating motion 

resultant velocity vector of combined wave and 
current motion 

velocity component parallel to the beach 



xx: 



dimensionless velocity parallel to the beach 

velocity component parallel to the beach at loca- 
tion of breaking waves 

mean velocity (across surf zone) parallel to beac: 
vertical velocity component 
fall velocity of sand grain 

horizontal coordinate perpendicular to the beach 
dimensionless distance from the shoreline 
width of the surf zone 

horizontal coordinate parallel to the beach 

vertical coordinate 

incident wave angle 

incident wave angle at breaking 

incident wave angle at wave generator 

incident wave angle in deep water 

bottom slope 

specific weight of fluid 

dimensionless suspension height 

kronecker delta 

energy dissipation function 

energy dissipation due to bottom friction 

vertical sediment mixing coefficient 

surf similarity parameter 

water surface elevation 

mean water surface elevation 

azumth angle of wave ray 

XX i 



®n ~ outward directed normal of shoreline 

ic = ratio between breaking wave height and the depth 
of water at breaking 

IC" = shoreline diffusivity value 

X = phase function 

A = linear spacial concentration in sediment-fluid 
mixture 

|i = dynamic viscosity of fluid 
^ei ~ dynamic -eddy viscosity of fluid 

V = kinematic viscosity of fluid 
C = roughness parameter 

= water particle excursion due to wave motion at bed 
P = density of fluid 
Pg = density of sediments 
a = local wave frequency 
^ij ~ shear stress 
'^hi " ^^^^^ stress at the bed 
(j) = dynamic friction angle 
*^'s ~ static friction angle = angle of repose for sediment 
$ = wave sheltering angle 

X = dimensionless coefficient dependent on wave and 
sediment parameters 

Y = velocity potential 
u = wave frequency 

= resultant angle of energy density at sheltered 
shoreline point 



xxii 



ub scripts 

b = breaker line or bed load 
h = bottom 

m = maximum (time wise) 
o = deep water 
s = suspended sediment 
w = wave 



xxiii 



Abstract of Dissertation.-Er>e^ exited to the Graduate Council 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 



LITTORAL SAND TRANSPORT ON BEACHES 

By 

Todd L. Walton, Jr. 
June 19 79 

Chairman: Dr. B-. A. Chris tensen 

Co-Chairman: Dr. T. Y. Chiu 

Major Department: Civil Engineering 

The transport of sand along beaches within the surf 
zone is considered in this dissertation. The driving forces 
for sand movement are considered to be the breaking water 
waves which transfer a part of their energy to the shoreline 
made up of noncohesive sandy material. 

The primary goal of this dissertation is to develop a 
model of sand transport using the excess momentum flux of a 
water waves approach as a driving force for the water cur- 
rents and an energistics approach to develop the sand trans- 
port created by the water currents. Both bed load and 
suspended load are considered in the model which is an 
outgrowth of the past work of M. S. Longuet-Higgins on 
"radiation" stresses within the surf zone and of R. A. 
Bagnold whose energistics approach to riverine sand transport 
is adhered to in the development of a similar model for 
littoral sand transport within the surf zone of sandy beaches. 

The equation for littoral sand transport developed in 
this dissertation has- a linear (in breaking wave angle) 

xxiv 



component' of suspeTid^d -s-ediment -tiraTisport , and a nonlinear 
(in breaking wave angle) component of bed load sediment 
transport. By using published experimental data, two unknown 
coefficients for the model are found. Existing data used in 
model correlation indicate that the model is reasonably 
valid. 

As a secondary goal of this dissertation, the model is 
used to develop sand t^^ansport values along the sandy beach 
portions of Florida. The computed values of sand transport 
are compared to existing estimates with reasonable results. 

Additionally, a new method of presentation of littoral 
sand transport computation results is suggested which pro- 
vides a means of evaluating stability/ instability features 
of shorelines. An example is given of how bimodel wave 
climates of elongated bays may lead to the numerous insta- 
bility features (giant cusps) seen in these bays. 

Additional analytical modeling is proposed using linear 
formulation (in wave angle) of the sand transport equation 
and is found to be useful in predicting the shoreline shapes 
of many coastal features. 



XXV 



CHAPTER I 
INTRODUCTION 

1 . 1 Introductory Note 
Sandy Beaches have long enjoyed great popularity with 
both the residents of coastal areas and the enormous influx 
of tourists se-eking out these vacation meccas . It has long 
been recognized that these same beaches, an increasing 
source of income for coastal zone citizens, are in serious 
trouble due to erosion. Preservation of these beaches is 
not only desirable aesthetically, but is also an economic 
necessity. 

The economic significance of the erosion problem can be 
seen (at least in one state) from costs shared between the 
State of Florida and the Federal Government in order to 
preserve Florida beaches. As of 1970, estimated first costs 
of authorized Federal beach improvment projects in Florida 
amounted to over 76 million dollars for 108 miles of beach. 
Estimated first costs to correct all the existing erosion 
problems in Florida (includes authorized and unauthorized 
projects) amounted to over 113 million dollars for 2 09 miles 
of ocean shoreline (1). 

Of approximately 1000 miles of sandy beaches in Florida 
(see Figure 1.1), the annual quantity of erosion in the 



1 



2 



GEOR Gl A 



C Ferrcndina 
. ; M^^st. Marys River 



St. Aview's 
Inlet 



JockscvTville 
~' John's River 

St. Ajgustine Inlet 
Matanzas Inlet 



Daytona 




pQuca Da Lecn Inist 



Cops Curm-eral 



LEGEND 

ANMJAL SHCRajNE RFCE3SCN 

~~ 1 20 Ft. Plus 

iO-20R. 
5-)0Ft 
SH8 2-5 Ff. 
« I-3F1 
I Q-l Ft. 



C5 

Q 



Sedcsfian Inlet 

Ff. Fieme Inlet 

St. Luds Iniei 
Jupitar Island 
J'jpifBr Inlst 
rbi.71 53ccij Inist 
■3cynk>n iniet 
3cca .Raicn Inlet 

P^. E'.'iinjkzlss In (at 
BalKSrs Hciiiovfr Inlet 
Micmi 



0 



Scale 

20 50 Milfl 
.J i_ 



Key Vtet^ 



0 20 60 KiiciT^tars 



Figure 1.1. Erosion Situation in Florida 



3 



nearshore area has been estimated at 15,000,000 cu. yds. per 
year, with over 20% of the beach shoreline in a critical 
state of erosion. Factors that influence beach erosion are 



LITTORAL SAND TRANSPORT ON BEACHES 



By 

TODD L. WALTON JR. 



A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF 
THE UNIVERSITY OF FLORIDA 
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 



1+ 




1910 !92D 1930 1940 1950 I960 1970 



TIME (Yecrs) 

Figure 1.2. Relative Rise in Sea Level In Florida (from Hicks ' (.2)) 



5 



Dunea 



Beach 




Sea Level Aft a* Rise 
Inttid .Sear Level 



Initial Bottom Profile [ 
Hypothetical Bottom Profile After Rise in Seo Lave! 



Actual Bottom Profile After 
Rise Of Sea Lav^ And The 
C^jcaTtftafivft Balance Bet»*eeR 
Erosion And Bottom Deposits 



After Sea Level Rlsa 



Limiting Depth Between Predominant 
Neorsftoref And Offshore Material Littoral 
Drift Characteristics 



Figure 1.3. Bruun's Concept of Beacli Profile Response 
to Sea Level Rise (from Bruun, P. (3)) 



6 



volume of sand from the. .upper, heaxih profile will slough off 
to maintain an equilibrium bottom profile offshore. This 
hypothesis was first discussed by Bruun (3) who developed a 
quantitative relationship for the rate of shoreline retreat 
in terms of the rate of sea level rise which, for Florida, 
amounts to about 1 to 3 feet of beach recession per year. 
As many areas of the coast appear to be relatively stable 
(i.e. much less than 1 foot -erosion per year in the short 
term), it is reasonable to postulate that either the hypoth- 
esis is wrong or that a trend of shoreline erosion in response 
to rising sea level is not gradual but rather takes place 
during more severe wave activity such as occurs during 
hurricanes or extra-tropical storms. In areas of normally 
low wave energy, rapid response to high wave events does 
occur and lends credulity to the above theory. Severe 
erosion occurs on our shorelines during storms and complete 
recovery of the lost sand is never made to the beaches. An 
example of what can happen during a hurricane is shown in 
Figure l,i+ from the work of Hayes (U). This figure docu- 
ments the potential for a large storm (Hurricane Carla) to 
transport tremendous quantities of material offshore out of 
the zone of all but extreme wave activity where it will not 
be returned to the beaches. In this storm, over 100 feet of 
dune system was virtually destroyed and a sand layer varying 
in thickness from 1 to 9 cm was deposited out to depths of 



7 



20 Ft. 



Post-Caria Gi 
Bed Thickness 
l->9cm 

I— S-9cm 

I— 3- 6cm 

1- I -3cm- 
m - Sand Layer 
(Post-Carla) 

2--^-Ma\or Hurricane 
■~" Channel 




GULF OF MEXICO 



Artifical 
Channel 
— Hypothetical 
Density Current 
Flow Pattern 



Contour Interval 
0 5 



20 R. 



Figure 1.4. 



Sediments Deposited off Central Padre Island 
During and After Hurricane Carla on 
a Mud Bottom (from Hayes (4)) 



8 



120 feet by a combination of hurricane waves and currents. 
Figure 1.5 presents the results of a more recent study of 
the beach erosion occurring during Hurricane Eloise which 
made landfall just west of Panama City, Florida in September 
1975 (5). The dune erosion profile shown is the result of a 
composite profile taken over a 20-mile stretch of beach from 
Panama City Beach to Destin, Florida. Over 7 5 feet of large 
15 to 20-foot primary dune system was virtually destroyed by 
this storm. It will take years before part of the dune 
system will be rebuilt naturally from the sand deposited 
offshore during this storm, and from experience gained from 
other hurricanes the beach-dune system will never recover 
completely. 

Sea level rise coupled with severe wave events is not 
the only reason for our erosion problem though. Another 
major contribution to erosion is our inlet systems and their 
corresponding navigation channels either natural or artifi- 
cially cut through the littoral zone. In the sandy beach 
littoral zone of Florida alone there are 5 7 inlets. Fourteen 
of these inlets have Federally authorized navigation projects 
with authorized channel depths extending to the ocean or 
gulf of over 20 feet. There are at least 14- more navigation 
projects with authorized channel depths of 10 feet or more. 
For comparative purposes, natural controlling depths on the 
outer bars of "unimproved" inlets are on the order of 6 to 8 



9 




£=: 
0 
d) 

3 

+J 

O) ^ — ~ 
CQ 

tn 

■ 1- (O 

o s- 

r— 3 
LU Q- 
S- 

cD :r 
c: Q. 

o -a 

•1- E 

i- « 
i_ 

3 3 



O) 

4-' £ 
1- O 

<c s- 

-c — - 
c 

(13 ra 
T3 
CD -i- 
S_ S- 

o o ■ 

M- ■— 

<D U. 
CD 

n 

•1- +J 

to 
O <D 

CL. 

-a 



3 



u ca 

(D fO 



CO 



s- 

c 

(1) 
cs 



N0I1VA313. 



10 



feet. Many of our Feder^al. navigafion projects are not 
natural channels but have been cut directly through the 
corresponding barrier islands such as St, Lucie Inlet on the 
lower east coast of Florida. The effects of this inlet will 
be mentioned later. When a channel is either cut through a 
barrier island or dredged below the natural existing depths, 
the flow of water through the channel to the bay (or lagoon) 
on flood tide and to the oceanCor gulf) on ebb tide is 
increased leading to an increased capability of the channel 
to flush sand to its inner bay system or outer shoal system. 
The channel also acts as a barrier to littoral sand moving 
along the coast which eventually works its way into the 
channel and then migrates to either the inner or outer 
shoals of the inlet. The sand in the interior of the bay 
systems cannot work its way back out as there is no wave 
activity to agitate the- sediments into suspension such that 
the water can carry it out. Thus, the bay shoals of these 
inlet systems act as net sinks to the beach sand system. 
Volumes of material residing in the inner and outer shoal 
systems have been shown to be substantial (6). An example 
of the amount of sand stored in Florida's outer shoal systems 
is shown in Table 1.1. In Florida, considering present 
erosion rates, it has been estimated that over 200 years 
worth of sand resides in the outer shoals of these inlets 
(6). The sand moves to these inlets by means of littoral 



11 



O >, 

U > 

CO U 

>H en 





H 






i 










•ri 






H 


u 


fn 




O 


0) 


fC 


to 


> 


■H 


m 






m 




u 






u 


rt) 


dJ 




Q} 


>i 


■P 




+J 




tU 


(T3 




O 


H 


0 


o 


•H 








X) 


O 


CO 


c 




H 




M 


O 


Ct3 


4h 






O 


O 











w 










rH 


M 










T3 


n3 






0) 




T3 








rH 




•H 




m 




rH 




s 


^ 


CO 


M 


•H 








to 


CO 


fH 


0) 








03 


C3 


H 


c 


ra 




Oh 


1 


e 






CO 




< 


M 










1 






m 


a 


C 


M 




<U 


(U 




•H 


W 




3: 


■H 


o 


rH 


n} 




O 


U 


"-^ 


eq 






J 











O >! 

> 

>i 00 



m 

rH 

o; fO w 

+-> FQ X! 
fH 

U (T3 
(U > 

3 



o 
> 

IT) 

13 S 

■H H 

it3 (tJ 

r-l O O 



o o 



o 

•rH 



0) 



OCNCSICNJ-J-^;:f^aDlOlDa3rH 
LnuOLrjLOLOLnmLOLOLOi-ni-nLniD 

HiHHf-i'HHrHHrHrHrHrHrHH 

o 

•CP 
H 



OOOOOOOOOOOOOO 

--IrHrHrHHHrHrHr-IHrH'-HrHrH 



LD ID O 

to J- CO 

cn o- CD <n 

H H rH 



UD ID CD <£) 
O O O O 
rH rH rH rH 



M 

to 
o 
a 

14h 
H 
3 
O 



O O CO 



o 

CO T3 



■H to 

(0 CO 

O to 

/3 Oh 
bO 

C S 

O (P 

J 2; 



X 


X 


X 




X 


X 


X 


X 




X 


X 


X 


X 


CO 


CO 


CD 


O 


o 




CD 


O) 




:3- 


CD 


CO 


o 


CO 


CO 


CX) 


CD 


o 


CO 


CN 






cn 


00 


ij- 


o 


00 


o 


O 


CO 


en 


CN 




en 




O 


O 


i-H 


en 


H 
























CS 










rH 


































CO 










CO 








CO 


















(0 






M 


CO 






CO 


















CO 


rO 






to 










CO 




CO 




tO 


(U 






tO 


> 








CG 


to 


to 


+J 






CO 


CO 


eu 


o 




CO 




to 


+-> 


to 


0) 




0) 


to 


CO 




>-H 




CO 


CO 


Oi 


o 


a, 


rH 


(0 




10 


to 


to 


— ' 




lO 


to 




CO 




C 


H 


c 


0- 




o 






CL, 


to 


o 


lO 


+-I 


1— 1 


H 


to 






rH 


CO 








a 




.c 




•rH 




to 






CO 




CO 






to 


bO 


OJ 




O 


> 


to 


to 


to 




c 




to 


CO 


•H 


o 


lO 




•H 


•rH 


CJ 


Oh 




•rH 


0 






c 


•rH 


CX, 


to 


•H 








>, 


bo 








CO 


CJ 


a- 


X) 


bO 




tu 


bO 


u 


bO 


•H 






to 


O 


to 




•H 






•rH 


0 


•H 


pa 




> 




PP 


o 


Pi 


P3 


2; 




3: 


t3) 


m 



ac3 



to 
O 



in 


J- 




CD 






c- 


CO 


O 


en 


IT) 




un 


LO 


CN 


CO 




:t 


cn 


CD 




CD 


CD 


CD 


CD 


CD 


CD 


r-H 


rH 






rH 


rH 


H 


rH 


rH 


1 


1 

CO 




1 

CO 


J- 










CO 


Ln 


Ln 


cn 


LT) 










CD 


CD 




CD 


CD 










rH 


H 


rH 


r-H 


rH 











rH 

J- 
IJI 



CD CD CD CO 



rH CO CO 

J- d- CO 
CD CD CD CD 
rH H H H 
I 

or 
H 



CD UO CD CD 



o 


O 


O 


O 


O 


O 


o 


o 


o 


O 


O 


o 


O 


rH 


H 


rH 




rH 


rH 


rH 


rH 


>H 


r-{ 


rH 


rH 


H 


X 


X 


X 


X 


X 


X 


X 


X 


X 


X 


X 


X 


X 












r- 


CD 


o 


o 


o 


CD 






o 


CN 


CN 


o 


O 


cn 


CM 


CD 


CD 


CD 


CO 


o 


CD 




























CD 


00 


O 


CD 


O 


O 


O 


J- 




rH 


CN 


H 


ID 


CO 


lO 


CD 


O 


tD 














IT) 


H 



> 

•H 

Pi 



- (U 



CO >i 

to fn 

O to 

+J • 

CO -H 

rO CO 
W 



> 
•H 

Td oi 
c 

3 

O 



o 00 



tn 
o 



0) 

o 
c 
to 

!^ 

c 



■p -p 

0) CD 

rH rH 

C C 

M M 

tU c 

c o 

•H a) 

-P 

CO 

3 0) 

bO Td 
3 

< OJ 

o 

. c 

-P o 

CO 0-c 



> 

o 

rH 

3 

to 



0) CO 

HJ U 

•H 0) 

3 to 



H-- O 

CO 

to ^^ 

O to 
o n: 

to 

rH rH 

3 O 
C3 O 

to 

CO 

D. 0) 
CU P-. 

3 



to 

CO 

to 



•p 

to 

u ^ 

c 



rH 

>, C 
to 3 

m i-o 

to CJ 
P4 o 



t3 
C 

to 

rH 

CO 



bO 
O 
Q 

CO ' 
CO 

rO to CO 
0-1 CO CO 
to lO 
C PU 0-1 
10 

■H -P P 

TO CO to 
C <U lO 
M 3: W 



12 







o 


o 


O 


o 


o 


o 


o 


o 


ncl 




o 


o 


o 


o 


o 


o 


o 


o 






o 


o 


o 


o 


o 


o 


o 


o 


tn 




o 


o 


a 


o 


o 


o 


o 


o 


CD 


CO 


o 


CO 


t>- 


CD 


IT- 


CO 


o 


o 


O 




to 


rH 


oo 


CO 






tH 





0 






cn 






CO 


CO 


CO 




■rH 


cu 


cu 




cu 


(U 


0) 


(D 




H (D 


iH 


rH 




iH 


H 


rH 


rH 




(U H 


•H 


•H 




•H 


•H 


•H 


•H 






e 


S 






s 


e 


s 




O -H 




















CO 


CO 




CM 


CO 




o 




00 














CM 
















H 








o o 


o 


o 


O 


o 


o 


o 


(U 


H 


o o 


o 


o 


o 


o 


o 


o 




H '-^ 


o o 


o 


o 


o 


o 


o 


o 




•HCO 


















IH t3 


CN o 


o 


o 


o 


o 


CO 


o 


o 




CO CO 


LO 


un 


o 




CD 


o 


> 


M_l w 


CD CO 




r-- 


CO 


CO 




o 




0 


*^ 






#\ 












CM 






CM 














oo 


r-i 


cn 


<T> 




rH 




■P 


CD CD 


CD 


r- 


i>- 


CD 


CO 


LO 




C 


cn en 




cn 


cn 


CD 


CT) 


CD 


O 


H CU 


H rH 


iH 


iH 


H 


H 


rH 


rH 


tu 


rH 6 

•H (U 




!h 


>, 


>, 


>i 








Ph o 


•H O 


(U 






H 


rH 


0) 




rtJ 




^ S 












rH 


Cm 












s 






< S 


cu 


















CJ 










> 








<13 










O 








Q 



























o 








m 






Q) 
•n 








Fl 




•H 


O 












ft 














ft 










ne 




■H 

CO 


p 








>, 




CO 


G 








03 




•H 


CU 








0 




CO 


S 








CO 


OJ 


CO 




o 


0 


03 


•H 


rH 


•H 


CO 






rH 


PQ 


Ph 




■H 


2: 




ti-. 
















>, 






3 








0) 


OJ 


>, 


O 




in 


rH 


>4 


G 


P 




0 


G 


03 




m 


G 




fl3 


rd 




>, 


H 




iH 


OJ 


H 




<u 


CO 


0 


O 


CQ 


Is 


av 




M 


0 


c 








03 


<U 


G 


o 








•H 




0 


■H 


-H 






G 




C^l 


P 


rH 


•H 




■H 


CO 


•H 


(13 


0 


P 




W) 


03 


!^ 


O 




G 


ft 


f-i 


0) 


U 


O 




d 


03 


•H 




03 


rJ 


0 






> 


E-i 


a:; 



13 



sand transport along the beaches ; the primary driving force 
being wave activity. Figure l.S shows an example of inner 
shoal sand trapping for St. Lucie Inlet which was cut in 
1892 through the barrier island. As noted by the solid line 
in Figure 1.6, the total sand deposited in the inlet over 
the years 1892 to 1930 amounts to over 9 million cubic yards 
of sand. 

It appears £rom the trexid shown in this figure that 
inlets shoal rapidly in their early years and eventually 
reach an equilibrium inner shoal area (7). The trend for 
outer bar shoals of inlets is not so apparent. Figure 1.7 
shows a relationship proposed by Walton (8, 9) between the 
volume of sand stored in the outer shoal of an inlet and the 
size of the inlet as determined by its cross sectional 
throat area (which is related in turn to the volume of water 
flowing through the inlet). Three curves are given express- 
ing various severity of wave activity at the inlet. The 
following sand storage equations postulated by regression 
analysis by Walton (8, 9) are given for the three wave 
conditions : 

1 28 

V = 33.1 * heavily exposed inlets 

1 2 8 

¥ = 40.7 A^ * moderately exposed inlets 
1 2 8 

V = 45.7 A^ ■ mildly exposed inlets 
where ¥ = volume of sand stored in the outer bar in cubic 
yards, and A^ = cross sectional area of the inlet in feet 
squared . 



14 



cn 



g 

cn 
o 

CL 
hi 

o 



< 

cn 



(0 

a 



cn 

GO 
00 



^ 2 



220- 
200 
180 
160 
140 
120 
100 

80 

60 

40 

20 



cn 

CD 

"S 

— a 

cj 
cri 

<s 

cn 

a 
CD 

CO 

CD 




pTrend Line for Shoaling Pattern 
in St. Lucie Inlet. ( From 2 miles 
North of In let to I mile South of 
niet in I.C.W.W.) 



I — Trend Line for Shoaling Pattern 
n St Lucie Inlet. (Vicinity of 
3/4 mile Radius of Inlet.) 



1883 18s: 1900 1910 1920 1930 1940 1950 1960 



YEAR 



Figure 1.5. Deposition of Sand in the Interior of St. 

Lucie Inlet (from Dean and Walton (7)) 



15 



X . — r— 

03 • O) U- U- • . 

r— . (/) fa 

Lu X - - - trt — • 

• H— brt ui vl (/I ^ f-™ 

OJ • - m (O tn 

CC'rtaO- Q- .— 1/1- 

C3 (/» * -C fO O > O U *TJ 

OJ O tC i/lt/»i— J3 (Oi/lC 

•TJ > C >,£rrt3*J CT»Q_C 



' LU 1/1 U 



-a c -o 
c; > 3 



«e -a u "O oj oi-c £ c o 




























ro 




m 




O 




fT3 








2 












O 








u 


> 


W 




(/I 




o o 




a: j3 




u 








c 




_2 


>> 








tj 










u. 










5 












a 




o 


t- 


o 








o 


< 


=3 




d 



n3 <: 



OJ U1 

a o 
> :e u 



I 

■in s 
o 



o 

"CVJ 



— o 



— tti 



a 
o 

cn 



8 

-o 
c 
a 























(J 












































z 


LL. 










u. 
















U. 


■ -3 












O 










. lO • 


-3 












x: 


3; 


u. 








c 










u . 


c 




•o 


■ u. 




o 
















c 






OJ 










01 1 — 








- cn • 












> 








O 


















— 






^ 1 — LLi 


cn 


a) 


^ 






cx 


« - 








i. o 


UJ o 














"c 




O U OJ c 






Ol 


c: 


a. 


c 


^ ii 


<: 


-5 


(3 


^ 13 O 


-4-> 


at 


O) 














cn 


3 w cn 




u 


c 


CI -4-> 




O — 






(/I 


iTJ C 0) 


OJ 


c 






(/> "3 


I. a. 




■t-i 


rt5 


0) 0) — S- 




o 


(H 


i. 


ta 


<z 


ra 3 








a 1 o 


ts 


a. 












m 




in 


yD r-». o ai 


a 















— OJ "B 

c 

_ o 



i 

O 
O 



i 4 i i 

O O ifJ <M 

(>4 ~ 



o poo o o lo 

o 



crt 
O 

c: CO 

rO 

OJ c 

cn o 

S_ f— 

O (t3 

on 

E 
s- o 
« i. 
cn u. 

s_ 

O) CO 
■M 4-3 

o ■— 

<u 

3 O 
4-J 

O) n3 

s- 

1/1 cC 
Q. 

I — 

c o 

O -r- 

•r- -M 

-P O 

(O O) 

1— (/5 
OJ 



0) 

3 



16 



As seen -in the . aLove. equations j the more severe the 
wave activity on the outer bar, the smaller its storage 
capacity is — i.e. wave activity limits the size of the outer 
shoal area by driving shoal material back to the beaches. 
The size of the inlet is the main controlling factor though 
as larger inlets store more sand. Thus deepening an inlet 
and consequent enlargement of an inlet's tidal prism may 
well cause additional shoaling on the outer bar if bar 
storage equilibrium has not been reached. The sand necessary 
to make up this additional storage volume must come from 
adjacent beaches. 

Figure 1.1 is a view of the erosion situation along 
Florida's coast in 196 3, but is similar to the erosion 
situation today. The large black spikes where the critical 
shoreline recession exists are at locations of inlets thus 
confirming our expectations that inlets cause erosion to 
adjacent beaches. Of course, it is also well known that in 
areas of a predominant net sand transport along the beaches, 
improvements such as jetties at inlets can cut off the 
natural flow of sand and thereby starve downdrift beaches of 
sand. In Florida, on practically every south side of an 
inlet (downdrift side), erosion is excessively high. It is 
not uncommon in Florida to have large stretches of shoreline 
adjacent to inlets undergoing recession at rates upwards of 
10 feet per year. 



17 



Another cause of "apparent" erosion to our shorelines 
is that of barrier overwash. Many of our low barrier islands 
are very susceptible to wave action occurring over the 
barrier island during periods of high tides with a consequent 
driving of sand into the bay systems as overwash fans. Very 
little is known as to the quantities of sand .lost during 
overwash events (10). 

/ Causes of erosion are then due to the two modes of sand 
loss; offshore sand lossybrought about by the onshore- 
offshore sand movement mode,^and longshore sand loss (which 
moves sand toward inlet traps) brought about by the "littoral" 
sand transport occurring due to wave activity on sandy 
beaches. Erosion due to both modes of motion can be de- 
scribed by a simple continuity of sand equation and applied 
to predict shoreline changes. The topic of this thesis 
though will only deal with the longshore sand transport, 
"littoral" drift, which appears as the major cause of erosion 
when man interferes with the beach -_( 11-2 3 ) . 

1 . 3 Ob j ectives 
This thesis will specifically deal with a predictive 
model for determining sand transport along sandy beaches as 
caused by water wave activity. Although in the vicinity of 
inlets or estuaries other factors such as tidal currents 
come into play, only the natural forcing function of gravity 
water waves will be dealt with in this thesis. The model 



18 



will be applied to the jnovement .O-f. cohesionless material 
(sand). Using the predictive model, sand transport will 
calculated for Florida's shoreline and a linearized sand 
transport model will be considered for use in predictive 
techniques for shoreline changes. 



CHAPTER II 
REVIEW OF SURF ZONE MODELS 



2 . 1 Review of Existing Longshore Sand Transport Models 
Numerous longshore sand transport models exist in the 
literature although none give better than an "order of 
magnitude" agi^eement with exisiring field and laboratory 
data. As the present models are mostly based on wave param- 
eters as the driving forces, it is important to note that 
no sand transport would occur in most of the models if wave 
action were not present to both stir up the bottom sediment 
and drive the longshore current. Most of the more popular 
surf zone littoral sand transport models (24-38) are listed 
in Tables 2 . 1 and 2.2. 

Equation 2.1(a) was historically arrived at by attempts 
to relate the volume of sand transported along the coast to 
a wave power term referred to as the "longshore energy flux, 
P^. Early researchers made empirical formulations of the 
type: 

= Constant • pj (2.1) 

Caldwell (24) first found an expression based on data with 
an exponent n - 0.8. Other researchers using other data 
made separate attempts to refine this equation with various 
values of the constant and the exponent, n. Many of these 

19 



20 



Table 2.1. Littoral Sand Transport Models. 



Reference 



(2.1a) = K^^ (25) 

or 

(2.1L) = (26) 

where = Q^Y(Sg - 1) (l-p^) 

H J. o c , COS a, „ 

(2.2a) I = K, \ ^ ^ (27) 

s 2a w 
f s 

or 2 

YH C , cos CL _ 

(2.2b) I = ^ V (27) 

s 2b w 

^ 4-/3 
cos a, n /o tang sin a, 

(2.3) Q, = K3^ — ^ -^^ ^ (28) 

d 

(2.5) S=i, ti =— ±— .(B (^)^^.S -1)^ 

^ ^ ^ gd-fi- ) ^ "hwm ^ 

V ^ -l)d 

(2.6) = B*d ^ g= . exp -(-A* § ) (31) 

r^(— ) (1 + — — ) ) 
z 



21 



Table 2.1. Definitions. 



Kij = constant 

= volumetric rate of sand transport (1^/t) 
Pj2_ = longshore energy flux (~) 

- immersed weight sand transport rate A) 

t 

3 - beach slope angle 

= wave breaking angle 
w^ = fall velocity (1/t) 

= wave breaking height (1) 
Y = specific weight of fluid (m/£^t^) 
T = wave period (t) 
d = grain size (1) 

^^b' ^gb ^ "^"^^ speed at breaking (1/t), wave group spe 
at breaking 

^2 = Chezy friction factor 

= ripple parameter = shear of flat sand bed 

total shear 

A* = empirical coefficient ( = 5 BijkerOl}) 
= empirical coefficient ( = 0.27 Bijker CSU'^) 
C = roughness parameter ( = 0,045 C BiikerOl)) 
P = fluid density (^/l^) 
Pg = sediment density (^/l^) 
g = acceleration of gravity Cl/t^) 
^hwm ~ ^^^i^um bottom orbital velocity (1/t) 
Pq = porosity of sediment in place 
S = specific gravity of sediment 



22 



Table 2.1. Definitions (continued) 



K - constant related to ratio of wave breaking 
height to depth 

E = energy density of waves (5^) 

t^ 

1^ - xmmersed wexght transport rate per unit width 

(— ) 
t^' 

^s' " efficiency constants 
1, t, m = length, time, mass 



23 



0) 

o 
c 

u 

0) 

Pi 



ta 

C 
3 



C 







CO 






CO 




ID 


CN 


CO 


CO 


CO 


CO 


CO 


CO 


CN 








a 

-H 
















(fl 














H 


■P 


rH 












td 


d) 










H 




-p 


nj 


-P 












<u 




(U 








0) 


Q) 




to 












tiO 


n3 




rd 










IXJ 


6 




g 


o 






iH 


> 


•H 




•H 










03 


•n 


u 


■1 — 1 


!0 


w 




CJ 


CO 


f— 1 


l-H 




CO 




QJ 








ear 
















>i 














^ 
















-M 


Mh 


u 








>> 


>i 


C 


■H 




iTJ 


o 


O 


nJ 


IT) 


O 




0) 


<D 


(I) 


CD 


"O 
\ 


\ 


6 
\ 


^ 




>, 


w 


CO 


=& 




4h 




\ 


\ 


\ 

=rt: 










■P 
















& 
















03 
















(U 
















>. 
































Mh 










>i 


>, 




-H 


n3 


n3 










O 




Q) 




m 


IT) 




\ 


s 




>) 
-\ 


>> 






o 




CO 


CO 


CO 


CO 


CO 


\ 

CO 






H 


e 


e 


6 






>1 
































03 


O 


m 




CO 


o 


o 


o 


O 


H 


o 


O 






r-H 


H 



to 

H 



CO 
H 
O 

o 
o 



o 
o 

CN 













03 


CN 




CD 


CN 




UD 


CD 


CO 


o 


rH 


O 


TO 


O 


o 


o 


O 


o 


O 



o 
c 
<u 

Q) 
Lp 

OJ 
Pi 















C^ 








CO 




CO 






CO 


CO 


CN 






^ ' 










T) 












C 












to 


o 












w 










c 










IT! 


n) 


to 




IT) 


o 


> 


6 




OS 


e 




n) 


C 




w 


0 


w 


00 


M 




a 











o 




o 


CN 


CN 


d- 




00 


o 


O 


o 


o 


o 



a, 
to 

II 



Ok 

ti 



CO 

CO 

r-l 

a 
a 

•H 

«i 
a 
m 
e 

■H 

-d 
I 



H 



24 



formulas are sununarxzed iji . TaiJLe. .2 ^ 2 . Later CERC (Coastal 
Engineering Research Center) revised the formula with an 
exponent n = 1.0 which is now in use as the standard littoral 
sand transport equation used by the Corps of Engineers (25). 

Limitations in the model exist though due to its neglect 
of factors pertaining to the material transported such as 
grain size of sediment, density of sediment, porosity of 
sediment in place- and ■ -temperature effects which have been 
shown important in suspended load in rivers and can make a 
factor of two difference in the transported load where a 
wide variation in temperature exists (39). Another short- 
coming of this formula is that beach slope is neglected. 
The constant in equation 2.1(a) is dimensional. 

A modification of equation 2.1(a) was first made by 
Inman and Bagnold (M-0) changing the volumetric rate of 
sediment transport to an immersed weight sediment transport 
rate as in equation 2.1(b). This allows for a dimensionless 
constant (more physically appealing) as well as accounting 
for grain density and sediment material porosity effects. 

A plot of this equation with considerable laboratory 

and field data is shown in Figure 2 of Inman and Frautchy 

(25). The dimensionless coefficient K^^ of equation 2.1(b) 

was found (26) to be K-, , = 0.25. Other researchers have 

lb 

given various values of K.^^^ equal to 0.2 (32); 0.4 (37); 
0.77 (38); and 0.8 (25). Many of these constants are summa- 
rized in Table 2.1. 



25 



Longuet-Higgins (^.1) has . criticized the above formula- 
tion as the parameter called "longshore energy flux" is 
neither the longshore directed component of energy flux nor 
the onshore component of energy flux. As the model does not 
differentiate between bed load type of transport and sus- 
pended load transport, both modes of sediment motion are 
included in this model. 

Equation 2.2 was postula-ted in a heuristic model by 
Dean (27) on the assumption that sane transport moves as 
suspended load. This formula does take into account many of 
the missing factors of equations 2.1(a) and 2.1(b), such as 
beach slope and temperature (via grain fall velocity), as 
well as grain density and porosity of sediment. Additionally 
the model can be used for the prediction of sand transport 
when a longshore current other than that driven by waves is 
present. An additional parameter of importance in this 
model is that of a bottom friction factor, C^, which is 
dependent on inviscid flow bottom orbital amplitude and 
velocity. This factor will be mentioned again later. The 
critical shortcoming of this model is its inability to 
predict bed load which has been noted by others (42, 38, 43) 
to be a major portion of the total sediment transport under 
conditions of low wave steepness and large sediment grain 



sizes 



26 



Equation 2.3 was postulated by Iwagaki and Sawaragi 
(28) on the basis that the average littoral transport is 
proportional to the shear velocity of the mean longshore 
current. They use a Kalinske-Brown type of sediment trans- 
port formula. The mean longshore current used in the formu- 
lation was that of Putnam, Munk , and Traylor (44). Limita- 
tions in this formula are evident in the lack of grain 
density considerations, sedinaent porosity, temperature 
effects (due to the use of grain size rather than fall 
velocity) and the dimensional form of the proportionality 
constant. Density and porosity considerations can be taken 
into consideration by a simple transformation used in equa- 
tion 2.1(b). 

Equation 2.U, postulated by Le Mehaute and Brebner (29) 
on physical reasoning and empirical information, has the 
same limitations as the previous equation along with the 
absence of two important parameters, beach slope and grain 
size. 

Equation 2.5 was postulated by Thornton (30) using an 
energistics approach as in Bagnold (45) for calculating the 
sand transport per unit width within the surf zone. Thorn- 
ton's model takes into account most all of the important 
sediment transport parameters except for sediment porosity 

but does not define the efficiency factors B or S which 

s s 

makes the equation unusable in a practical sense. Addition- 
ally, Thornton correlates the work rate of bed load transport 



27 



to the total available energy flux of the waves rather than 
to the energy expended in the longshore direction. As a 
result, the energy expended in moving sand back and forth 
(in the onshore-offshore direction) is included within the 
available power for longshore sand transport, while, in 
reality, no work is being done in the longshore direction by 
this energy .dissipation. 

Equation 2.6 has been po^stulated by Bijker (31) for bed 
load transport in the surf zone based on extensive laboratory 
tests and a sand transport model proposed by Frijilink and 
modified by Bijker (31). Weaknesses in the model are ex- 
pressed in the form of empirical coefficients which were 
determined on the basis of laboratory tests and give values 
ranging over two orders of magnitude. Additionally beach 
slope and material porosity are not taken into consideration. 
The lack of existence between an expressed form of bed 
roughness (in the surf zone) and the Chezy friction factor 
adds a qualitative factor to the model which makes it diffi- 
cult to apply in practical application. Bijker (31) proposes 
a suspended load as a function of the existing bed load. 

2.2 Review of Longshore Current Velocity Mode ls 
As most of the longshore sand transport formulas take 
into consideration the longshore current either explicitly 
or implicitly, a review of the longshore current models is 
pertinent. Most of the longshore current formulas predict 



28 



only the mean of the longshore current velocity with the 
exception of the Longuet-Higgins model (46), and the Bowen 
model (47). 

There are four basic approaches to the development of 
predictive equations for currents in the surf zone: conser- 
vation of mass, conservation of momentum, conservation of 
energy, and empirical correlation. Most of the more familiar 
equations, are presented in- Table 2. .3. An additional long- 
shore current model by Thornton (30) has not been included 
in the table but can be found in Reference (30). Some of 
the equations have been expressed in slightly different 
forms than originally published in order that they all 
incorporate the same parameters; however, these changes deal 
only with the geometric relation for the breaking depth and 
breaking wave height. The changes do not alter the currents 
predicted by the equations- Symbols, utilized in the equa- 
tions are listed in the preface. Much of the following 
review of these equations must be credited to Thornton (30). 
The equations will be reviewed in order of increasing analyt- 
ical usefulness with empirical equations having the least 
utility due to their lack of physical justification or 
rational. 

Empirical correlations — Two types of empirical equations 
have been developed. The first type employs physical reason- 
ing to determine the form and grouping of the important 



parameters which are then correlated with experimental data. 
Brebner and Kamphuis used both the energy and momentum 

equations to obtain reasonable groupings by dimensional 
analysis of the important parameters. Linear regression was 
then used to find the best fit for the longshore velocity to 
a large number of data that they had measured in the 
laboratory. 

The second, type of analysis employs multiple regression 
techniques. Sonu et al. (49) used this method to weigh the 
various independent variables collected in their field 
studies. They found that the most important variable affect- 
ing the mean longshore current velocity was the angle of 
wave incidence, and the second most important, although much 
less, was the wind. These results are conflicting with 
those of a similar analysis reported by Harrison and Krumbein 
(50) who, using data collected at Virginia Beach, Virginia, 
found the most important variable to be the wave period 
which proved to be insignificant in Sonu's analysis. Sonu 
also performed a multiple quasi-nonlinear regression analysis 
which showed the most important variable affecting the mean 
longshore current velocity to be the wave height. In a 
later study, Harrison (51), using another set of data col- 
lected at Virginia Beach, found the incident wave angle to 
be the most important, followed by the wave period, height, 
and beach slope, respectively. Harrison notes that the use 



30 



of empirical equations is limited in application to "similar" 
situations; it is not possible to extrapolate to different 
type beaches than considered in the empirical formulation. 
Another problem with using empirical formulations is that 
they are devoid of physical basis and, as such, can give 
spurious correlation and conflicting results .( 52 ) . 

Continuity considerations--Chiu (53) and Inman and 
Bagnold (40) .derived similar ..expressions using the continuity 
approach. These formulations are based on the fact that the 
incident waves introduce a mass flux of water into the surf 
zone which is then manifested as a spatial gradient in the 
longshore current. Both developments consider a plane beach 
of infinite length, implying that mass is uniformly intro- 
duced into the surf zone along the beach. The current will 
grow (since mass is continually being supplied to the surf 
zone), and at intervals, it is necessary that there be 
outflow from the surf zone unless the current becomes un- 
bounded. It is postulated that this outflow occurs in the 
form of rip currents which are evenly spaced along the 
coast- The equations thus contain an unknown parameter — the 
spacing of the rip currents. Unfortunately, few measurements 
have been made of rip current spacings so that the use of 
these equations requires additional experimental data. 

It should be noted that, due to the mass flux of waves, 
there is always transport of fluid into the surf zone and 



31 



-that,, in all the physical models .whether considering a mass, 
energy, or momentum approach, the mass flux must be accounted 
for in order to obtain a bounded solution. Thus, the assump- 
tion that the mass transport is uniformly returned across 
the surf zone, is at least implied in all the developments 
which do not include concentrated return flow by rip currents. 

Chiu and Bruun (53) also considered the case where rip 
currents are absent and the return flow is distributed 
uniformly over the vertical plane containing the breaker 
line. He reasoned that waves breaking at an angle to the 
beach contribute mass to the surf zone and locally raise the 
mean water level as the breaking wave crest propagates down 
the beach. This results in a slope of the water surface 
between crests which creates a longshore current. The 
longshore current is balanced by bottom shear stress related 
to the velocity through the Chezy formula. Galvin and 
Eagleson (54) reasoning from the continuity approach, 
equated a hypothetical mass flux across the breaker line 
proportional to the mass contained in the longshore current. 
Using both field and laboratory data, the two mass fluxes 
were correlated. 

Energy considerations--Putnam at al. (44) also derived 
a mean longshore current equation from energy considerations 
alone. The derivation equates the changes in energy flux to 
the frictional energy losses parallel to the beach. A 



32 

difficulty, with the resulting equation, is that it involves 
two undetermined constants, a friction factor f and the 
percentage of the wave energy available to the longshore 
current, "s," which makes the equation very difficult to 
apply. 

Momentum considerations— Putnam, Munk , and Traylor 

used both the energy and momentum equations to derive the 

first rational equations JLor^^jdescribing longshore currents . 

They considered the flux of mass and momentum into a control 

volume of differential length bounded by the breaker line 

and the shore. The change in momentum flux across the 

breakers directed parallel to shore is balanced by the 

bottom shear stress. Solitary wave theory was used to 

calculate the momentum of the breaking waves . In this 

manner, they obtained an expression for the mean longshore 

velocity related to the angle of wave incidence , breaking 

b 

wave height H^^ , bottom slope tanB, wave period T, and fric- 
tion factor f. Embodied in all of the momentum analyses is 
a friction factor that relates the velocity to the bed shear 
stress and represents an empirical coefficient. This equa- 
tion was subsequently revised by Inman and Quinn (52) who 
found that a better fit to the data originally collected b/ 
Putnam et al . , and additional field data collected by the 
authors, was obtained if the constant friction coefficient 
xn the original equation was changed to be a function of t le 
velocity. 



^33 



tn 

(0 

H 
3 
£ 

o 

tM 



c 

3 

u 

?^ 
o 

CO 

bo 

c 

o 



CO 

<N 

0) 
.-I 

n3 



C 

o 

-H 

m 
w 







CO 












tn 






•H 












•H 






CO 




CO 








CO 










•H 








>, 










CO 








H 


c 


T3 


to 




>i 




CO 




ro 


o 


0) 


c 


(U 


H 


CO 


•H 


0) 


C 


•H 


CO 


to 


CO 


CO 


to 


CO 


to 


to 


-H 


to 




to 


C 




>1 


to 




to 




e 




lO 






rQ 


e 


H 




3 


1 




1 


to 


1 


3 


3 


H 


+j 


.H 






C 


M 


4-' 


B 


to 


c 


ifl 


^ 


to 


to 


to 


G 


U 


O 


<u 


O 


!m 


o 




O 


(U 


o 


•H 


s 


•H 


0) 


-H 


0) 


■H 


e 






o 


u 


C 


u 






0 






g 


•H 




•H 


to 




e 




a, 










nb 








e 


c 


e 


c 


e 


e 


c 




to 


o 




o 


tj 


to 


[J 


o 



c 

!^ 
U 
3 



a isi 

O 3 

CO 

bO CO 
C CO 

o o 
J u 
o 

c < 

to 

tu 



CO 

o 

3 

< 



CN 



in 

C 
•H 

CO 



cn 

CM 



3 
O 
CO 



c 
•w 

CO 



o 



a 

•H 

CO 

o 

CO 



CO 



to 

3 
•H 

J2 

a< 

S 

to 

I 

fn 
0) 
C 

pa 



CO 
CM 



a 



O 1^ 

c 

a 
o 



(a 



co 



C 
•H 

to 



O 
4- 



C 
•H 

o 

H 



CO 



to 

D. 
S 

I 



(N 









o 


3 






H 


r- 




CO 


o 


o 




o 


1 




CQ 


+ 


C 




to 


Em 


4-' 


<X> 


CD 


rH 


O 


CO 


CO 


O 


o 


O 


o 


+ 


+■ 


















• 




O 





o 

CO 
•H 

u 



o 
H 

# 

CM 



,^3 

a 
a 

•H 

CO 

CO 
CM 

CM 



CM 
If) 



g 

o- 

I 

§ 

i 

M 



8 

CO 

o 
o 

rQ 

CO. 

C 
to 



CO 

* 

CO 

o 

H 



3& 



c 
o 

-H • 

+J O 

w 



o 

•H 

-M 
nj 

a 



I > 

-p 

c 

0) 

C 

3 O 

CJ Csl 

(U 4h 
O P 

w 

bO w 
C 
O 
J 



c 

(0 
0) 



o 

o 
< 



H 

to 
o 

•H 
U 
•H 

g 
I 

•H 

6 
0) 



ca 

M 

o 
o 

Xi 
3 
C 
•H 

m 



CN 



o 

•H 

■P 

> 

u 

to 
c 
o 

CJ 

m 
w 

n3 

s 



a 

CM 
C 
■H 

CO 
CO. 

C 

m 
+-> 



TO 
H 



c 
o 

E 

(U 
CO 
B 

o 
o 

to 
s 



8 
(M 

C 
•H 

CO 
CO. 

C 

m 



CM 



0} 
X) 
" 3 

G H 
O O 
•H C 

-P -H 

<TS 

> CO 
U +J 
OJ G 
CO a) 
G ^^ 



O 
CD 



3 
o 

CO 

CO CU 
CO 



Xi 

a 

CO 

o 

o 

00. 

G 
CO 

■p 



H 
CO 



0) 

" 3 
G H 
O 
C 
■H 



■P 
CO 
> 

u 

CD 



CO CD 

c u 



o 



3 

o 
CO 

CO O4 

(0 '-H 



3 

c 

•H 

CO 

s 

CO 

o 

o 



c 

CO 

■p 



CO 



H, 
CN 



CO 



a 

CN 

c 

•H 

CO 



OX 

+J 











C 






0 




G 


•H 




0 


■P 




•H 


CO 




-P 


> 




CO 


U 




> CO 


tu 


CO 




CO 


0) 


m > 


G 


> 


CO (0 


0 


lO 


G S 

0 


a 




0 >, 


s 


>i 


u 


3 


U 


>> CO 


4-' 


CO 


bjQ 4-' 


G 


■P 


U -H 


CU 


■H 


(U H 


B 


H 


C 0 


0 


0 


U C/D 







X\ 

a 
c 

•H 

CO 

X 

tiO 
00 
CN 

CN 



a 

CO 

o 
o 



CO. 
G 

CO 
■P 



CO 
CO 



o 

CN 



fa 



< |CN 



■X3 

0 

3' 

.5 

§ 
O 



era 



0 
H 



CO 

U 
o 

■p 

3 



O 



00 

CO 



G 
lO 
S 
G 



U 

to 
e 



10 



G 
O 
CO 
0) 
H 
bO 
to 

w 
i 

G 

•H 

> 

to 

C5 



CO 

in 



I 



3 



CO 



C 

3 

I 



3 

,G 
O 



o 



T3 

O 
G 
bfl 
lO 
P3 
I 

G 

to 

6 
G 
M 



0 
H 

i 

i 

a 
•p 
3 



O 

to 

I 

a; 

G 
3 

I 

E 
to 

G 
4-> 
3 
PL, 



CD 
H 

CM 



m 
m 
> 

c s 
o 

•rH O 

-(J -H 

> O 

P -H 

0) fj 

M (U 

C CU 



O 

a 
e 

c 

<D 

e 

o 



I 

u 

•H 

u 

-H 

s 

E 
>, 

CO 

< 



in 

in 



C 

o 
w 

H 
bO 
ITS 

w 



CM 



CO -H 

CO CO 

CO 



c 

o 

•H 

(0 



< 

e 

c 

s 
o 



PS s 



a; 












U 


0) 


s 










c 


CM 






3 


D 


o 


c 








O 


iSl 


•H 






C 






CO 






•rH 


(D 








a 


CO 




!^ 


a 




CM 


IM 


O 


;3 






c 


1 






•H 




•H 


CO 


CO 




CO 




CO 


^ — ' 


ba 


CO 


al 






CN 


c 


CO 


C 




. 


V 


o 


o 


•H 








J 


u 


CO 




CO 


4- 




o 










c 


< 




/3 


CN 


co 








V 




Me 




bO 








CO 


CO 


t= 


CD 








LO 





a- 



o 

03 



o 



CM 



CO 

CO -H 

CO CO 
<u 
u 

CO 



C 

o 

•H 

to 

■H 

to _ 

Pi S 



C 
< 

■p 
c 

0) 

e 
o 



a 

CO 

o 

o 

a 
c 
•w 

CO 

a 

c 

to 

4-1 
iiclM-( 



to 



03 

c 

•H 

tin 

m 

•H 

a: 
I 

<u 

3 

§ 
h3 



36 



Eagleson (55), usi.iig.the same cojitroJ. volume approach, 
developed a mathematical model to represent the growth of a 
longshore current downstream of a barrier,. In the associated 
laboratory experiments , it was found that a large percentage 
of the fluid composing plunging breakers (most common labora- 
tory breaker type) is extracted from the surf zone. This 
fluid already has a longshore velocity to which is added the 
longshore component of the breaking wave. This argument 
provides a mechanism for growth of the longshore current 
downstream. The asymptotic solution to the differential 
equation showed that the system is stable and that the 
growth of the currents was bounded. These results agreed 
qualitatively with laboratory results and demonstrated that 
unless there are perturbations inducing gradients in the 
wave energy in the longshore direction, the current system 
tends to be uniform alongshore and stable for stationary 
wave conditions . 

Bowen (M-7) used a conservation of momentum approach to 
determine the longshore velocity distribution across the 
surf zone for the case of a plane beach with a constant 
lateral shear stress. Reasonable results were obtained when 
compared to laboratory data. In the case of no lateral 
shear stress, his results are comparable to those of Longuet- 
Higgins except for inclusion of set-up. 



37 



Longuet-Higgins ' (-46 independently of -Bowen, derived 
a longshore velocity distribution across the surf zone for 
the case of a plane beach based on "radiation stress" theory. 
Reasonable results are also found for this theory when 
lateral mixing effects are included. As "radiation stress" 
theory will be used in the development of a sand transport 
equation, this model will be expanded on later. 

In -the- late 1960's Galvin^(56) critically reviewed 
the mean longshore current theories and tested those that 
predicted longshore current distribution across the surf 
zone as well as mean longshore current agreement. Galvin 
used both latoratory as well as field data. Galvin concluded 
that the most applicable equation appeared to be that of 
Longuet-Higgins (M-S) when proper unknown constant coeffi- 
cients were used. if 

Sonu et al. (49) 'Conducted field experiments and found 
poor correlation when compared to six of the above equations. 
Their experiments did point out the importance of the near- 
shore topography on the current system, and how this may 
affect the outcome of such results. Shepard and Sayner (57) 
pointed out that another reason for lack of agreement between 
mean longshore current' theory and field experiments could be 
the variation of current across the surf zone; each field 
data point is usually based on only a single location in the 
surf zone and additionally consists of effects based on a 
spectrum of waves rather than one frequency component. 



38 



Many simplifying assumptions are necessary in developing 
the theories. Since exact expressions are not available, it 
is necessary, inside the surf zone, to select approximate 
expressions for the wave speed, wave shape, water particle 
velocity, partitioning of energy in the wave field, a long- 
shore variation of waves and currents , and velocity and 
energy distributions across the surf zone. It is possible 
that -improved theories for longshore currents will require a 
better understanding of the highly nonlinear waves in the 
vicinity of the surf zone. 

The importance of considering factors other than wave 
parameters, such as the wind, was demonstrated by Sonu's 
empirical correlations. He found the wind to be the second 
most important variable in his set of field data. This 
shows the difficulty in comparing tests, particularly field 
data where information concerning the effects of bottom 
topography and winds often is not included. The extrapola- 
tion of data from one particular location to another without 
accounting for the importance of these effects can lead to 
invalid results. 

All of the equations involve unknown coefficients to be 
determined experimentally. Generally, the friction factor 
in the momentum and energy equations is evaluated in the 
same manner as in open channel hydraulics. The validity of 
utilizing results from steady flow situations without inertia 



39 



considerations certainly needs to be investigated further 
and could hopefully result in a refinement. 

Sonu points out that another possible improvement might 
derive from consideration of the dynamic processes of energy 
dissipation in the surf zone environment. The difficulty is 
that the flux of energy used in generating longshore currents 
is only a small fraction of the total available energy, and, 
as such, represents a second ord.er phenomenon (56). A 
difficulty with using the continuity approach is that, 
although it allows a description of the mean current, no 
procedure has yet been developed, based on continuity consid- 
erations, which can provide a prediction of the variation of 
the current across the surf zone. 

Momentum approaches remain the most useful approaches 
to date (55) when dealing with longshore currents and will 
be used in the development of a sand transport model. 

2 . 3 Review of Friction Factor 

The longshore current in many of the above formulations 

is dependent on bed shear stress which in turn is dependent 

on bottom friction factor f ' . The determination of the bed 

w 

shear stress and hence friction factor for uniform steady 
flow has been fairly well established. For oscillatory 
flow, and particularly combined waves and currents, the bed 
shear stress is not well formulated. This is due primarily 
to the lack of good data. 



40 



Both -field and laboratory. data exist for the evalua- 
tion of the friction factor "f" although field data have 
been obtained on an "energy" type approach whereas most 
laboratory results have been attained by other means. As 
the two approaches provide somewhat different answers 
both will be discussed. 

2.3.1 Field Data 

Field data for friction factor f ' have been obtained 

w 

by measurement of wave heights at two different locations 

and by calculating the energy loss due to bottom energy 

dissipation by friction, then equating theoretical energy 

dissipation to measured energy dissipation. The friction 

factor relates the shear stress per unit area at the bed 

T, to the horizontal flow velocity above the bed u, as 
h hw 

^h = 1 ^^;p%w' ^2.21) 

other equations of the general form of equation 2.21 have 
appeared with the constant 1/2 replaced by 1 or 1/8. 

The rate of wave energy dissipation per unit area, E^, 
following Putnam and Johnson (58) is 



^f ^h ^hw ^2.22) 
Combining equations 2.21 and 2.22 yields: 

^f \ P%w' ^2.23) 



1+1 



All variables in the equations presented thus far indicate 
instantaneous values. For the sinusoidal flow at the bed, 
predicted by small amplitude theory. 

^hw = ^hwm • (2.24) 

where u^^^ is the maximum horizontal bottom particle velocity 
at the upper limit of the boundary layer (i.e., the maximum 
bottom velocity of inviscid f low) . The argument of the sine 
term includes the angular frequency, w = 2Tr/T, with T the 
wave period and the time, t. Substituting equation 2.214 in- 
to 2.2 3 and integrating over the wave period gives, 

^f = It P %wm^ • (2.25) 

Equation 2.25 is the average rate of wave energy dissipation 
per unit area of ocean bed due to the frictional shear 
stresses produced at the bed. 

Bretschneider and Reid (59) give a solution to the wave 
energy conservation equation presented in the form of a wave 
height reduction factor, K^. This factor accounts for 
bottom friction losses in a way similar to linear wave the- 
ory refraction and shoaling coefficient treatment of wave 
energy transformation. Combined they show the change in wave 
height due to three effects: 

H = ^s ^o • (2.26) 

The coefficients of K^, K^, and are the friction, refrac- 
tion and shoaling coefficients. 



42 



Br.etschneider (6 0) .using Jield data from wage gages 
on two oil platforms in the Gulf of Mexico calculated fric- 
tion factors averaging 0.10 6 by first calculating the 
factor from field measurements and then comparing the 
so obtained with the postulated energy dissipation using 

the Bretschneider and Reid (59) solution to obtain f. 

w 

In analyzing bottom losses for waves generated in 
shallow water, Bretschneider (5Q) found an average friction 
factor of f^ = 0.02. Iwagaki and Kakinuma (51) used signifi- 
cant wave heights at two stations off the coast of Akita, 
with the Bretschneider and Reid (50) method, to estimate 
friction factors ranging from 0.066 to 0.180, with an average 
of 0.114. Iwagaki and Kakinuma (62) found average friction 
factors for 4 other coasts of 0.280, 0.130, 0.110 and 0.100. 

The results of various field tests (60-63) are given 

in Table 2.4, which lists average f values as well as the 
' ^ w 

range of friction factors measured " in the study. ■ Unfortu- 
nately, the results of field tests for friction factors are 
seen to vary widely which is due to a number of reasons: 
(1) additional wind energy effects on wave height in the 
measured values of wave height (which could increase or 
decrease wave height depending on whether wave energy was 
added or extracted from waves by wind between measuring 
points); (2) wave frictional damping due to viscid bottom 
fluid such as encountered on mud or fine silt bottoms; 



1+3 



Table 2.i+. Results of Field Measurements of Friction Factor. 



Reference 



Location 



f Average 
w 



Range of f ' 



Bret Schneider 
(50) 



Gulf of Mexico 



0.106 



0. 060-1. 93U 



Bretschneider 
(60) 

Iwagaki 

and 
Kakinuma 

(61-D2) 



Kishi 

(63) 



Gulf of Mexico 



Akita Coast, Japan 
Izumisano Coast 
Hiezu Coast (1963) 
Nishikinohama Coast 
Hiezu Coast (196i4) 
Takahama Coast 

depths 20-M-O ft. 
slopes - 0.006 

Niigata , Japan 

depth 6-8 ft. 
slope 0.018 



0.02 



0.116 
0.280 
0.156 
1.100 
0.094 
0.100 



0.035 



0.066-0,180 

0.054-0.260 
0.560-2.320 

0.020-0. mo 

0.060-0.160 



0.03 -0.04 



41+ 

(3) incorrect measurement, .refraction between measuring 
points; (4) no dependence on bottom roughness. 

The second reason for field data scatter has been 
mentioned by Bretschneider and Reid (50) as causing friction 
factors an order of magnitude larger than might be experi- 
enced otherwise in areas with mud bottoms. 

The fourth reason mentioned, that bottom roughness was 
not taken into- account is contrary to what one would expect 
as shown in steady flow friction factors from pipe flow 
data (64). 

2.3.2 Laboratory Data 

Jonsson (65) compiled available laboratory data and 
conducted additional experiments dealing with turbulent 
boundary layers in oscillatory flow. From this study, he 
developed a classification of the flow regimes similar to 
that for steady flow. The classification is based on a 
roughness parameter and characteristic Reynolds number. The 
wave boundary layer in nature can always be considered in 
the "Hydraulically rough" turbulent regime according to this 
classification scheme. ' 

Unlike the boundary layer in open channel flow, which 
essentially extends over the entire depth of flow, the 
boundary layer under wave motion constitutes only a very 
small fraction of the vertical velocity distribution. This 
is because the boundary layer does not have an opportunity 



45 



to develop under the unsteady velocity field of the wave 
action. Above the boundary layer, the free stream is well 
described by the potential flow theory (for small waves). 

Jonsson using an oscillating flow tunnel to simulate 
prototype wave conditions in the laboratory was able to 
measure the vertical velocity profile in the boundary layer 
and determine the bottom shear stress. He found that for 
simple harmonic wave motion .the velordty profile in the 
boundary layer could be approximated by a logarithmic dis- 
tribution and that the instantaneous bottom shear stress was 
related to the velocity by the quadratic shear stress 
formula 

■^h = ^w f ^hl^hl ^2.27) 

or 

^h = ^w f ^hwm ^°s(a3t-8) | coscot-S | (2.28) 

where ^-^vim '^'^^ velocity amplitude of the oscillating flow 

just above the boundary layer, 6 is a phase lag, and f is a 

w 

friction factor associated with the wave motion. 

Jonsson also found that the friction factor was practi- 
cally constant over an oscillation period. The constancy of 
the friction factor for particular flow conditions is an im- 
portant result which allows for a better determination of the 
combined shear stress due to waves and the longshore current. 
Using the available data from several sources, '.■Jonsson found 



■46 



that the wave friction factor for laminar wave boundary layer 
flow was 

f„ = — (2.29) 
ew 

or, in terms of the friction coefficient as defined in the 
energy dissipation approach used in the field studies, is 

f; = (2.30) 
W /R 

ew 

where R^^ = wave bottom amplitude Reynold's number 

= . (2.31) 

For the case of rough turbulent flow Jonsson found that the 
friction factor for simple harmonic wave motion could be 
represented by an equation of the form 

1 . 1 _ „ , . ^h 



_ + log — — = -0.08 + log — ~ (2.32) 

4 /f 4 Vf~ ^ 

w w 

where r is a measure of roughness, and E,^ is the maximum 
horizontal water particle excursion amplitude of the fluid 
motion at the bottom as predicted by linear wave theory 

^ 2 sinh kh (2.33) 
Equation 2.3 2 is based on the roughness parameter r 
being equivalent to the Nikuradse roughness parameter (65). 
As the roughness is a function of ripple height and spacing 

in wave flow, f is also a function of the wave 

w 



1+7 



characteristics. This is because, for the granular beds 

consisting of a particular grain size, the ripples adjust 

their dimensions according to the wave motion, and it is the 

ripple geometry that determines the effective roughness. 

Using additional laboratory data, others (66-69) have 

postulated results similar to Jonsson where the entire range 

of wave friction factor f depends on both the wave bottom 

w 

amplitude Reynold's number R^^ and boundary roughness — wave 

amplitude ratio. A summary of some laboratory results for 

the rough turbulent range are given in Table 2.5. 

Here it is important to again note that f^ and f^ are 

not necessarily equivalent as wave friction factor f^ has 

been arrived at assuming no phase lag between bottom wave 

stress and wave velocity while Jonsson 's data include a 

phase lag between maximum bottom wave shear and maximum 

bottom wave velocity. . Also, Jonsson 's postulated friction 

factor f takes into account '' ■^ bed roughness effects 
w 

whereas the data used in estimating the friction factor f^ 
did not account for bed roughness form explicitly. 

At present there is only a qualitative understanding 
of the ripple geometry as related to sand wave characteris- 
tics. Generally, the ripples are much more symmetrical in 
shape and much longer crested as compared to those found in 
alluvial channels. Inman (70) collected a large number of 
observations of ripple geometry and wave conditions from 



48 



Table 2.5. Results of Laboratory Measurements of Friction Factor. 





Tp=!t Pnnditions 




f 

w 




Jonsson 


Oscillating water channel 


0. 


02-0. 


30 


\03 } 


LLlxDUJ-cii L ijtju.iiu.cix _y xci_y t3X 

additional reanalysis of 
other laboratory experiments 








Tunstall 


Wave flume study 


0. 


09-0. 


50 


and 










Inman 


U, T 
2 t; R 








Iwagaki 


Wave Flume Study 


0. 


01-0. 


40 


and. 


laminar boundarv laver 








Tsuchiva 
(67) 










Kamphuis, J. W. 


Similar to Jonsson 


0. 


02-0. 


30 


(68) 


experiments 








Car St ens, et al. 


Oscillating flow in channel. 


0. 


01-0. 


50 


(69) 


turbulent boundary layer 









R = Ripple height. 



49 



Southern Calif orBia .beacJies - T±i£se ..otiser'vations extended 
from a depth of 170 feet to the shore. Since the wave and 
sand characteristics vary from deep water to the beach, the 
ripple geometry would be expected to vary also. These ob- 
servations showed that the size of the sand is the most 
important factor in determining the geometry of the ripple. 
In general, the coarser the sand, the larger the ripples. 
Also, there was a general correspondence of decreasing 
ripple height with decreasing water depth. The ripples were 
smallest in the surf zone where the higher orbital velocities 
of the waves tended to plane the ripples off ; the ripples 
were almost nonexistent for surf zones with fine sands. The 
ripple wave length was related to the wave bottom orbital 
excursion. As the bottom orbital velocity of the shoaling 
waves increased, the ripple length decreased, increasing the 
effective roughness, but, at the same time, the ripple 
height is decreased, decreasing the effective roughness. 

Carstens et al. (69) in oscillatory flow laboratory 
experiments found that the ratio of ripple amplitude to mean 
sediment diameter and the ratio of dune amplitude to dune 
wave length were found to be unique functions of a single 
variable — ratio of water motion amplitude to mean sediment 
diameter. If the amplitude-to-diameter ratio is less than 
775, the ripples are two dimensional with essentially 
straight and level crests and troughs. If the amplitude-to- 
diameter ratio is greater than 1700, the bed is flat 



50 



regardless of the initial condition. On the intermediate 
range from 775 to 1700, the ripples are three-dimensional 
with ill defined crests and troughs. Ripple amplitude was 
found to decrease almost linearly with increasing water 
motion amplitude in the range of three dimensional ripples. 



CHAPTER III 

REVIEW OF EXISTING DATA ON LONGSHORE SAND TRANSPORT 



3 . 1 Introductory Remarks 

Existing littoral sand transport data fall into two 
categories, laboratory data and field data. Data on sand 
transport and waves are diffiault to take under laboratory 
conditions, not to mention the inherent difficulties in 
taking field data. The following review of available data 
on sand transport includes manner of testing where available 
data exist, and conditions of such testing. The review is 
comprehensive to the best of the author's knowledge, although 
the data are by no means ideal. In almost all tests of sand 
transport many important parameters have not been measured 
and very rarely has differentiation been made between bed 
load transport and suspended load transport. Much of the 
data herein referred to have been used in past studies for 
analysis of sand transport models although only order of 
magnitude agreement has been found to date. Many of the 
laboratory and field tests included here have also been 
reviewed by Das (71). 

3 . 2 Laboratory Studies 

A number of laboratory experiments have been conducted 
to determine the mechanisms causing sand transport in the 



51 



52 



surf zone. This discussion is limited to the three- 
dimensional studies simulating conditions in the prototype. 
Krumbein (72) conducted one of the first of these experiments 
and concluded that the mean littoral drift was a function of 
the deep water wave steepness, H^/L^. Subsequent studies by 
Saville (73) showed that a maximum transport occurred for a 
wave steepness of 0.025, and, for steepnesses greater or 
less than this value, the transport was less. Saville also 
found that under low wave steepness much of the sand trans- 
port occurred on the foreshore of the beach as bed load 
while for higher steepness waves much of the sand transport 
occurred as suspended load. Both Saville (73) and Savage 
(7M-) concluded that greater than 9 0% of all longshore sand 
transport due to waves occurred within the surf zone (land- 
ward of breaking waves). This has also been confirmed by at 
least one researcher in field studies (75). 

Although laboratory results imply that the transport is 
a function of wave steepness, the reason for this dependence 
has not been established. Galvin (76) conducted a series of 
experiments of breaking waves on laboratory beaches . He 
developed a classification for determining whether the waves 
develop into plunging or spilling breakers as related to 
wave steepness and the beach slope. These results, when 
compared to the littoral drift studies in the laboratory, 
indicate that maximum transport occurs for a plunging-type 



53 



breaker and that the rate of. transport may be more a func- 
tion of the manner in which the waves break than the wave 
steepness. Battjes (7 7) has defined the parameter which 
controls the type of wave breaking as a surf similarity 

parameter, c, = "^ ^"^ , where B is beach slope angle, H is 

o 

wave height, and is deepwater wave length. The smaller 
the surf similarity parameter the more the wave breaking 
process approaches spilling breaker conditions. 

The following Table 3.1 provides a review of laboratory 
experiments which will be referred to in later discussion. 
Table 3.2 gives the range of wave basins used. 

The angles are the nominal settings of the wave 
generators to the beach. In the case of "snake" wave 
generators, angle is controlled by wave phasing. 

A short review of each series of tests follows: 

3.2.1 Krumbein Laboratory Study (72) 

The laboratory study by Krumbein was conducted in 19142 
at the University of California, Berkeley. Longshore trans- 
port rate, longshore current velocity, and wave and sediment 
characteristics were measured. The maximum rate of removal 
of sand from the updrift end of the beach was determined 
from the rate of feed of material to a hopper. The rate was 
adjusted to the capability of the waves to move sand. 
Longshore currents were measured by using floats and confetti. 



O 



M 
+J 
M 
• 0) 
O E-i 



f4 

O 
+-> 

u 
a 
a 

> 



to 

•H 

-P 
•H 

C 
l-l 



0) 

o 

rH 



ban 
a <u 
<u 

o u 
bO <D 

c o 
< ^ 



-p 

a d) 

0) 

CO O 



■P -P 

•H ip 

0) 



M 

C 
O 

o 

0) 



O 
•H 



p 
c 



ra 

o ji, 
p 

CO 
■H 

., 

•H <D 
t3 -P 

oj o e 

CO rO g 

to O 

a 



to 

Q 

o 

<n 
o 
u 

3 
O 

to 



to O 
• o 

o 



CO 



O 

rH 



o 

rH 

H 



O 



O O 
rH <M 



O 
CO 



rH 
H 



















T3 






!>. 








<D 


■P 




H 




OJ 


<U 










C 




•H 




rH 


rH 




rH 






0) 




j:: 




<U 








a 


o 


o 


— , 


o 




A:: 








o 


c 


c 




u 




f< 


?^ 






03 


0) 


•H 


a; 


•H 




0) 






0) 


C 


Sh 


> 


w 


to 




m 


m 




pq 








o 














Jo 




pu 










>, 




>1 




>, 


to 


>) 


c 


















u 


to 




O -H 


O 




o 


G 


o 




o 






+-> 0) 


■P 


< — i 


■p 


to 


p 




■p 


0) 




to 


to 


rH 


to 




rO 


to 


to 


bO 




!^ S 


u 




u 


>, 


U 


> 


1^ 


to 




O 3 


o 


> 


o 


to 


O 




o 


> 




IB ^ 




to 


^ 


x: 




to 


X) 


to 




IB 




to 




(B 


CO 


to 


CO 




J 










J 











o 



CO 
CN 



O 
CO 



O 

I 

in 



o o 



o 
J- 



o 


CO 




CD 






CO 




rH 


in 


00 






jj- 






CO 




CO 


CN 






















H 


r-i 


rH 








CN 


H 


rH 




rH 




















r- 




o 


CO 








rH 


CO 


in 


CN 


rH 


CN 


rH 








CN 


CN 


CN 






















1 

CD 


1 

cn 


1 

CN 


1 

in 








1 

03 


1 

CO 


1 

CO 


r~- 




in 


CO 






H 


rH 


o 


H 


o 


o 


o 


o 




































o 


















=f 


o 


o 








in 


CO 


CO 


ID 


o 


in 


J- 


cr> 






























H 




CN 
1 


rH 


H 








CO 


H 






r- 


1 

=t- 


1 

lO 








in 


1 

in 


1 


1 


<T> 


H 


f- 








CN 


H 


o 


cn 


































rH 


rH 


rH 


rH 


in 


Ol 


<J> 


o 


o 


o 


in 


in 




in 




lO 


to 






rH 


CO 


CO 


(O 


CD 






















CN 


CN 


CN 




rH 


H 




rH 


rH 


OJ 


o 


o 


o 


o 


O 


O 


CN 


o 


o 




in 


CO 


CO 


in 


in 


o 




CO 


CO 






















c^- 


o 


o 


o 


o 


rH 


rH 


o 


o 


H 




y — s. 






















00 


CD 


cn 






o 


rH 


Ol 


CO 














CO 


T3 CO 


CO 


CO 



bO - 










C 


C 








•H • . 


o 




P" 




H 


Xi 




Mh 




rH 


CO 




H 




IB ■ 


■H 




OJ 




S rH 


J 




Q 






— ' 














>, 










!^ 




O QJ 


o 


o 


O 




P 


p 


H 


Pi 


U 


(0 (U 


to 


CU 


IB 


0) 


U O 


t< 


O 




A< 


0 -H 


o 




0 


•n 


!^ 


Xi 


rO 


Xi 


•rH 


to 


to 


cq 


to 


cq 


-3 











55 



Table 3.2. Summary of Wave Basin Dimensions. 



Source of Data 

Krimbein 
Saville 

Shay and Johnson 

Sauvage and Vincent 

Fairchild (CERC) 
and Savage 

Price and Tomlinson 

Barcelo ' 

Bijker 



Basin Dimensions 
(feet) 



58.2 X 38.7 X 2.0 
122 X 66 X 2.0 
122 X 56 X 2.0 

Not Available 

150 X 100 X 30 
(North Sector) 

200 X 150 X 30 
(South Sector) 

190 X 75 

64 X 32 



Approximate Beach 
Length (feet) 

38 
60 

60 (Counts 25-81) 
33 (Counts 82-123) 

Not Available 

30-98 (Variable) 

? 

32 

? 



56 



The maximum velocity parafllei to shoi^e near the plunge line 
was measured by releasing a soaked string, and recording the 
movement of the end of the string over a fixed time interval. 
Wave heights were measured with a combination point and hook 
gage in which the hook-could be set to the troughs of the 
waves and the point to the. crests. The wave height was then 
obtained by reading the difference in gage heights on a 
single vernier. The mechanical analysis of sand used in the 
study gave a median grain diameter of 0.5 mm, although the 
sand was not well sorted. Specific gravity of material 
(sand) was assumed equal to 2.65. Unfortunately wave breaker 
angles were not recorded. 

3.2.2 Saville Laboratory Study (73) 

Saville conducted a laboratory study at the University 
of California, Berkeley. The total transport rate due to 
bed load and suspended load was determined by installing a 
weighing device at the downcast end of the beach. The bed 
load rate was measured by installing hoppers on the beach. 
Breaker angles were determined from vertical photographs . 
The mechanical analysis of sand showed a median gran size 
dgg = 0.30 mm. The sand was relatively uniform in size 
with a specific gravity of 2.69. 

3.2.3 Shay and Johnson Laboratory Study (78) 

A laboratory study was conducted by Shay and Johnson 
at the University of California. VJave and sediment 



57 



characteristics, longshore transport rate, and maximum 
longshore current were measured in the tests. Wave height 
variability was observed in the experiments therefore many 
readings were taken during each run and an arithmetic average 
of the readings was used to characterize the wave height, 
^ave heights were measured with a point and hook gage as in 
Krumbein's tests. The bed load rate was measured by install- 
ing hoppers on the beach for small generator angles. These 
hoppers were removed when wave splitters were used for 
angles of 3 0 degrees or greater. Both Bed'load and total 
load were measured in these tests similar to those of Saville 

Longshore current velocities (maximum and the average) 
were measured by releasing fluorescein. The breaker angles 
were determined from vertical photographs as in Saville 's 
study . 

Equilibrium beach profiles were taken after a stable 
condition was reached by inserting a piece of sheet metal in 
the beach perpendicular to the beach contours and tracing 
the sand profile under water with a grease marker, with a 
view to study the influence of wave steepness on beach 
profile . 

The median grain size was d^^ = 0.30 mm and specific 
gravity was 2.69. 



58 



3.2.4 Sauvage and Vincent.- T. a, boj-a-tory Study (79) 

Two series of laboratory experiments were conducted by 
the authors at the hydraulics laboratory at Grenoble, France. 
In the first series of tests, the beach was set at an angle 
of 15 degrees with the wave makers. In the second series, 
the angle was varied from 5 degrees to 70 degrees. In 
addition, there was a unique set of experiments in which 
three types of sediments with different characteristics were 
used. The volume of material, fed from a distributor and 
transported along the beach, was collected in a trap at the 
downdrift end. The data from the measurements of longshore 
transport and the wave characteristics are presented ' only in 
graphical form in the paper by Sauvage and Vincent (79). 
The actual measured data are not available. 

In the studies by Sauvage and Vincent (79), the mean 
grain sizes of the three materials used were 0.5 mm, 1.5 
mm, and 1.0 mm, and their respective specific gravities were 
given as 2.60, 1.40, and 1.1. 

3.2.5 Savage and Fairchild Laboratory Study (74, 80) 

In a laboratory program on the studies of longshore 
transport at the former Beach Erosion Board (BEB), presently 
known as Coastal Engineering Research Center (CERC), several 
tests were conducted from 19 5 8 to 1965. Ten of these tests 
were reported by Savage (74), and all of the tests in edited 
form appear in Fairchild (80). 



59 



The transport studies were made in the north and south 
sectors of the Shore Processes Test Basin (SPTB) of CERC. A 
detailed description of the SPTB, the sand transporting 
system, the sand traps and the sand weighing system is ..in 
BEB Technical Memorandum No. 11M-, 19 59. The total quantity 
of longshore drift caught in the sand traps was recorded for 
every 5-hour interval during the test. 

To establish a r e la tionsiip . between the longshore 
energy and the longshore transport rate, the transport rate 
between the 20th and 30th hours of test was used, whenever 
the test continued over 20 hours. 

Due to the large variability in wave heights observed 
in the SPTB during the test, it was difficult to characterize 
each test with a particular measured incident wave height. 
It was decided to characterize the waves by a half-size 
Froude model in another wave flume (75 feet by 1.5 feet by 
2.0 foot deep) at CERC. By varying the eccentricity and the 
period in the wave flume, the wave heights (before visible 
reflection occurred) were measured. The average of these 
wave heights was considered as the height of incident wave 
at that particular depth of water. The wave height and 
eccentricity were derived from half-scale, Froude-model wave 
tank studies. 

A family of curves was obtained for wave heights as a 
function of wave period with eccentricity as a parameter. 



60 



These curves were used to diajpacterize the incident waves 
in the SPTB (8), and these values (due to the large vari- 
ability in wave heights in the measurements of wave heights 
in SPTB) were used to compute the wave energy in the SPTB. 
Fluorescein was used in most tests to measure longshore 
currents. Beach samples were also obtained in the tests.' 
Water temperature measurements were made during each test 
at frequent intervals. 

In most of the studies conducted at BEB, no training 
walls were placed in the wave basin to conform to the wave 
orthogonal for uniform distribution of energy in the beach. 
The BEB study also differs from other studies in such aspects 
as basin geometry, beach slope, wave height and period 
variability, and sand feeder height with respect to still- 
water level. 

Median grain size. was d^^ = 0.22 mm and specific 
gravity 2.65 (beach sand). The edited data appearing in 
Reference (80) are the results of the averages of various 
tests . 

3.2.6 Price, Tomlinson, and Willis Laboratory Study (81) 

Studies were conducted at Wallingford Research Station 
(England) on the effect of groins on a beach stable for a 
particular wave condition and beach material. Longshore 
transport was also measured on two occasions without groins. 

The wave basin is 190 feet by 75 feet equipped with a 
snake type wave maker, and tide and tidal-current generators. 



61 



The initial beach slope was. 1 on 17 .at .mean tide level. 
The mean grain size and the specific gravity of crushed 
coal used in Price, Tomlinson and Willis's studies (81) 
were 0.80 mm and 1.35 respectively. 

3.2.7 Barcelo Laboratory Study (82) 

Studies were conducted at the Portugese National Engi- 
neering Laboratory on the effect of groins on a beach for 
various wave conditions . Longshore transport was measured 
in the first hour of the tests by measurement of accretion 
volume at a long groin within the basin. Wave heights 
measured were an average of random wave heights which- cor- 
responded to a Gaussian distribution. Median grain size 
was 1.30 mm and specific gravity equal to 1.6 7 for pumice 
stone, the material used in testing. 

As data were not listed in (82) except in graphical 
form, wave energy values and sand transport values have to 
be taken off graphical plots in the report. 

3.2.8 Bijker Laboratory Study (83) 

The raw data presented in the Bijker (83) report are 
sketchy and considerable questionable aspects of the study 
are not answered. The longshore current induced in the 
model is not driven by waves alone but by a combination of 
waves and an imposed gravity slope in the longshore direction. 
As a result, the data do not pertain to wave driven sand 



62 



transport alone but must be the result of additional forcing 
functions. 

3 . 3 Field Studies 

The measurement of littoral drift along a particular 
beach and the meaningful correlation with the wave properties 
are an extremely difficult field task. Thus, only limited 
field data are available. 

The earliest published information on the distribution 
of sand transport along the beach profile for prototype 
conditions was by the Beach Erosion Board (SH). Unfortu- 
nately, comprehensive wave data wezre not obtained. The 
distribution of longshore currents and suspended sand, 
obtained from water samples, was measured from piers extend- 
ing across the surf zone. These measurements showed that 
the greatest sand transport occurred at the breaker line, 
where the turbulence was a maximum, and decreased shoreward 
with another peak in the swash zone — another area of high 
turbulence. Seaward of the breakers the sand transport 
decreased with increasing depth. There have been few other 
field experiments of this type. 

Improvements in tracer techniques, particularly using 
either radio-active or fluorescent sand tracers, have in- 
creased the intensity of littoral drift studies in the 
field. It is generally assumed that the fluorescent tracer 
moves as bed load. A great number of studies have been 



63 



conducted in recent years on fluorescent tracing and have 
been sununarized in a book by Ingle (75). Ingle also con- 
ducted a number of studies using fluorescent tracers on 
several Southern California Beaches. The results of these 
studies and some studies conducted by Thorton (30), and also 
earlier investigators, are all generally similar to that 
given by Zenkovitch (85) and shown in Figure 3.1, describing 
the variation of sand transport across the surf zone. The 
fluorescent sand grains are found in greatest concentrations 
along points of high turbulence such as wave breaking areas . 
In a bar- trough profile, the sand moves predominantly along 
the bar or in the swash zone. There is a minimum of tracer 
transport in the trough. During the tests shown in Figure 
3.1 the waves were relatively high and spilling breakers 
prevailed . 

Tracer studies have been conducted using both the 
Eulerian and Lagrangian approaches. Most investigators have 
used the Lagrangian approach in which the tracer is intro- 
duced at a particular location, and the concentration dis- 
tribution of the tracer is determined by obtaining samples 
at various sample points. The concentration of tracers is 
then determined by counting the tracer grains in the sand 
samples. The Eulerian approach is to sample in time along a 
particular line across the surf zone, traversing the path of 
the tracers. A stable platform or other work facility is 



64 




Outer Breaker Line 



Q 28 



4 

2 
0 





75 (m) ICO 



150^-4- 



X 

i— 
a. 

UJ 



Figure 3.1. Distribution of Longshore Velocity and Sediment Transport 

Across the Surf Zone (after Zenkovitch (85")) 



8 5 

generally required in this method. This was the approach 
used by Zenkovitch (85), working from a tramway traversing 
the surf zone. Bruun and Battjes (86) and Thornton (30) 
worked from a pier. 

The inherent difficulty of fluorescent tracer studies 
is that quantitative measurements require recovery of most 
of the tracer. Unfortunately, the recovery level is gener- 
ally very low, amounting to only. a few percent. This re- 
quires that accompanying measurements of the quantity of 
sand in suspension, or moving on the bed, also be determined. 
However, as a tool, or aid, for solving engineering problems 
in which qualitative information can be extremely important, 
fluorescent tracer techniques alone can be of great value. 
Due to lack of field data such studies are necessary for a 
large scale working sand transport correlation until much 
more field data are obtained. 

Table 3.3 presents a summary of field tests in which 
measurements were made of both sand transport and wave 
parameters. None of the field tests have differentiated 
between bed load and suspended load. 

A short summary of each of the various field tests 
follows . 

3.3.1 Watt Field Study (87) 

By measuring wave characteristics, longshore currents, 
and amount of material pumped by the bypassing plang on the 



66 



m 

>, c 

bO o 



c 
u 

> 
m 
:2 



3 
o 



rd 
a 





CO 




o 




1 t 
+-* 




Q 








J— M 




rH 






CD 


■H 


bO U 


!^ 




0) 


bO ^ 






a; 


0) bO 


M -P 




(11 CO 




bo fij 




O 




0) t3 




> C 






Cu m 
1 1 




E- S 


1 1 


n 





CO 
(0 



H 
<l) 







CO 


CO 










to 


on 








<u 


0) 


•H 






■p 


p 


p 


P 


P 


CO 


•H 


•H 


•H 


10 


•H 


tu 


CO 




CO 


> 

u 


(7D 


-3-1- 


T) 


T) 


T3 




T3 


6 


CU 


<U 


CL) 


CO 


tu 


•H 


P 


P 


P 




P 


P 


C 


fl 


C 


o 


C 


Cd 


0) 


0) 


(U 






<1) 


B 


s 




rH 


s 






3 




to 


3 


H 


u 








?j 


to 


p 


P 


P 


CO 


P 


3 


CO 


CO 


to 


•H 


to 


to 


In 


In 


In 


> 


In 


Vi 



3 
C7< p 



O 
CO 

c 

to 

CO E-< 



•H 

c 
o 



CO 

c 

.H to 
to C/0 

< 



c 

o 

•H 

P 

to 

u 

o 

bO 
c 

•H 

CO bO 



to 
to 



>, o 
^ p 

•H 

T) C 
C O 
to S 



P 

to 

c 

o 

•H 

to 

O 

0) 

c 

o 

•H 

P 

(U 

u 



O 0) 

O T-l 



t 

p 

CO 

(U 
O 

to 







































CO 
p 






ip 




c 






o 




me 






(1) 


tu 


0) 






<u 


p 


CO 






H 


(0 

s 


3 ft 
CO to 






C 




to u 






•H 


fO 






3 




<U 


6 


p 


P 


C 




T3 


? 


CO 


o 




to 


o 




•w 




to o 






p 


tu 


O 


bO 




ID 




H 




o 




o 


t3 


P 


to 


O 






•H 




o 


CO 


0) ^ 


ft 




< 









bo 
an 



e 



o 

tn 



P 

CO 

0) 
E-> 

rH 

•H 







O 




in 






in 










CD 






CD 






CN 




CM 






CN 






O 




o 






o 
















CD 






















o 




O 






O 








to 
































c 














y — ^ 


















O 


CD 






tn 






cn 


ip 


CD 






CO 








■H 






(0 


^ — ' 








rH 






•H 




p 


to 




to 




x: 


G 










O 




o 






o 


-H 








to 


O 




3: 












ip 






O 








eq 


•H 




0) 


rH 




to 






H 










m 


iH 


O 


to 




lo 








rH 


C 


O 










e 


cu 


0) 








p 


CO 


•H 










x: 


(U 


P 


o; 


X) 


0 


to 


to 


p 


H 


P 


,a 


.H 




•i—i 


e 


3 


C 


to 


0) 


to 




lO 


o 


o 


IH 




c 


O 








CO 






< 




M 







LO 

CD 



in 

<D 



Ln 
iH 

o 



CO 
to CO 
•H w 

c 

U 

o 



CD 
CO 



in 

CD 



O 



in 
CD 



o 

CO 



o 
o 



to 

COo 

lOtxi 



t3 


ip 








(U 






c 




•H 








CQ 






O 


to 


H 




tu 










CO 




to 




e 




(0 






ft 


P 


O 




<u 




C 




C 


e 


CO 






c 




•H 


to 


O 


o 








3 




13 




P 




U 


x: 








C 


■H 


c 


£-< 




a 


(0 




a 


to 




u 




> 


to 


6 


P 




c 


O 


o 


<U 


H 


1) 


O 


!^ 


U 


u 


rH 




ft 


•H 


eq 




O 


CD 


0) 


tM 


E-r 


10 


CO 






Oh 










o 



67 



north side of the north jetty at South Lake Worth Inlet, 
Florida, Watts attempted to relate the volume of longshore 
transport reaching the pump intake to the wave energy reach- 
ing adjacent shores. (87). 

In the South Lake Worth Inlet region of the Florida 
coast, net longshore transport is from north to south. The 
material intercepted by the north jetty of the inlet was 
bypassed froni the north side to the south shore of the south 
jetty to prevent shoaling inside the inlet and to nourish 
the south beach. The southerly component of longshore 
transport near the inlet was measured from a detention 
basin, which was prepared for this study. The sand was 
pumped into this basin. During the period sand could not be 
pumped into the basin, the pumping rate of the plant was 
used to estimate the material pumped. 

During the period of study, the pumping plant bypassed 
almost all the littoral drift moving alongshore inside the 
surf zone. Therefore, it was assumed that the pumped volume 
would represent the total southerly longshore transport rate 
in the nearshore zone. The material pumped into the deten- 
tion basin was periodically surveyed for measurement of the 
quantity. During the period the detention basin was being 
cleared and leveled, the material could not be pumped into 
the basin, but the material pumped during these intervals 
was estimated from the average pumping rate of 7 6.2 cubic 



68 



yards per hour, computed from the log of pumping time between 
January 1949 and December 1951, furnished by the Palm Beach 
County Engineer. Wave heights and periods were measured by 
a pressure gage located at Palm Beach Pier, 11 miles north 
of the inlet and in about 17 feet of water below mean sea 
level. The recording mechanism was programmed to obtain a 
12-minute record every 4 hours from 6 March to 10 June 19 52. 

Wave directions were measured twice daily by the use of 
a sighting bar and auxiliary sights attached to an ordinary 
engineer transit located on the roof of the Ambassador Hotel 
about 3.5 miles north of the bypassing plant. 

Significant wave heights and periods were computed from 
the wave records from a pressure gage in 17 feet of water. 
Frequencies of wave heights and periods were plotted. A 
wave direction frequency plot indicated that 7 5 percent of 
the direction were from north of east. The predominant 
direction of longshore transport along the Florida Coast is 
generally from north to south. Therefore, for the data 
presented in the analysis, the alongshore component:- of wave 
energy was computed for each month for southerly wave direc- 
tions (a<90°) from durations for each a and the corresponding 
recorded wave height and period data. The total monthly 
southerly longshore transport was evaluated from the pumped 
material. The southerly component of wave work was expressed 
in footpounds per day per foot of wave crest, and the 



69 



southerly rate of littoral transport wa.s expressed in 
cubic yards per day. 

Longshore currents inside the breaker zone were 
measured twice daily by using fluorescein at four locations 
at distances of 1/2 mile, 2 miles, 5 miles, and 7 miles 
north of the bypassing plant at the same time^-the wave 
directions were measured. 

Sediment 'Samples were taken during the study and size 
distributions were made by Emery Settling Velocity Tube. 
This material was a sand and shell mixture. The data for 
sampling stations 1/2 mile north of inlet and 1,000 feet 
south of inlet on different dates and at high, mean, and 
low tide lines, and at 3-foot water depth were presented. 

3.3.2 Caldwell Field Study (88) 

From measurements of the rate of a longshore sand 
movement of a beach fill placed at Surf side, south of 
Anaheim Bay, California, and the associated wave charac- 
teristics, an attempt was made to correlate the two (88). 
The beach profile changes shown by seven surveys out to the 
20-foot contour were used to compute the volume changes of 
beach fill along the shore. 

Wave heights and periods were obtained from 8-minute 
wave records taken at 4--hour intervals by wave gages in- 
stalled on the Huntington Beach Pier about 6 miles south of 
Anaheim Bay. The depth of water at the gaging station was 



70 



20 feet below mean lower low. water. . Utilizing meterological 
data, wave forecasting techniques were used to "hindcast" 
significant wave heights, significant periods, and the 
directions of wave approach. Hindcast heights and periods 
were used to supplement recorded heights and periods where 
gaps occurred in the record. Aerial photographs of the study 
area were also taken. Wave direction and period were also 
determined from the photographs. Hindcast ing and wave refrac- 
tion analysis were used in combination to determine the wave 
direction associated with each wave observation. The median 
grain size in the beach fill area was 0.42 mm. 

3.3.3 CERC Field Study (89) 

A study was made by the Coastal Engineering Research 
Center of the sand deposited behind an offshore breakwater 
at Port Huneme, California (89). The sand deposition rate 
was estimated by means of detailed hydrographic surveys . 
Wave height, period, and direction data were taken by visual 
observation at the site and correlated with the sand deposited 
in the lee of the breakwater. Unfortunately visual observa- 
tions were not available throughout the entire length of 
the study. 

3.3.U Moore and Cole Field Study (90) 

A transport rate of 4,6 80 cubic yards per day was 
measured near Cape Thompson Alaska, from the growth of a 
sand spit during a 3-hour period (90). The material was 



71 



deposited on the spit by waves 5..- feet high with a period 
of 5.5 seconds and a 25-degree angle to the beach. The' 
median grain size was 1.00 mm. 

3.3.5 Komar Field Study (38) 

Studies were conducted at El Moreno Beach located on 
the northwest shore of Gulf of California, in Baja California, 
Mexico, and at Silver Strand Beach near San Diego, California 
(38). The purpose was to obtain field measurements of the 
bed load transport rate over short periods of time and, 
simultaneously, to measure the waves and currents to be 
able to test relationships between the longshore transport 
rate and the wave energy flux. 

The measurements of longshore transport rate of sand 
were made through use of natural sand colored with a thin 
coating of fluorescent dye. The sand advection rate was 
determined from the time history of the movement of the 
center of gravity of the sand tracer which had been intro- 
duced onto the beach. The thickness of the sand in motion 
was obtained from the depth of burial of the tracer sand in 
cores of the beach face. The product of the advection rate 
times the cross section of the sand in motion gave the sand 
transport rate. The wave direction and wave energy flux 
were obtained from simultaneous measurement-: of wave charac- 
teristics by an array of digital wave sensors placed in and 
near the surf zone. The wave sensors were of pressure type 



72 



measuring pressure variation near the bottom. The energy 
density obtained from measurements of the pressure trans- 
ducers was corrected for the damping effect due to over- 
lying depth of water using the linear pressure response 
factor. The root-mean square wave height obtained from the 
energy density, and the characteristic wave period obtained 
from the frequency spectrum were used to compute the wave 
energy flux at the breaker depth. Median grain size was 

drn = 0-6 mm at El Moreno Beach and 0.17 5 at Silver Strand 
5 0 

Beach, Specific gravity was 2.65. 

3.3.6 Thornton Study (30) 

In this study bed load measurements were made of sand 
transport across the surf zone (30). The traps extended 
8-10 inches off the bottom and were estimated to have an 
efficiency of 50-90%. The portion of suspended sand 
captured by the traps is unknown; therefore, it is questionable 
as to whether the entire amount of sand measured in these 
experiments is bed load or whether an important percentage 
is also suspended load. Wave measurements were made at an 
instrumented pier. 



CHAPTER IV 
SAND TRANSPORT MODEL DEVELOPMENT 



4 . 1 Introduction 

Past investigations, as discussed in Chapter 2,' have 
attempted to relate the mean longshore currents to wave- 
induced momentum, energy, or. mas.S--£liix into the surf zone. 
The distribution of longshore currents can be similarly 
investigated by considering the changes in the momentum, 
energy, or mass flux across the surf zone as has been done 
by Longuet-Higgins (46), Bowen (47), and Thornton (30). The 
present analysis utilizes the momentum principle and the 
concept of "radiation stress," the excess momentum flux due 
to wave motion (91), to develop the working equations for 
longshore current and available power for sand transport. 

It is known that, due to the fluctuating water particle 
motion of the waves, there is a momentum flux component. If 
the waves have a direction component parallel to shore, a 
longshore current can be generated due to changes in the 
longshore momentum flux component of the shoaling waves 
(46). It is also known that there must be a displacement of 
the mean water surface elevation to balance the changes in 
the onshore momentum flux component of the shoaling waves 
(92) . 



73 



7^ 



In waves, the momentum flux is the sum of the pressure 
and the product of two velocities. It can be shown that the 
average momentum flux is nonlinear in wave height. In order 
to specify the excess momentum flux of the waves, it there- 
fore becomes necessary to consider nonlinear, or higher 
order effects of the wave motion. Since the waves are 
assumed in the present model to be the mechanism responsible 
for the generation of the longshore currents , much of the 
following discussion will be devoted to describing the 
waves . 

The treatment of periodic gravity waves is generally 
developed by perturbation schemes built on the exact solution 
for the linearized equations of motion with specified bound- 
ary conditions. Difficulty, with this type of analysis, is 
encountered when considering higher order theory with non- 
horizontal bottom boundary. For waves traveling in water of 
varying depth, two different approaches can be employed: 
the analytical method or the energy flux method. The analyt- 
ical method involves solving the boundary value problem 
accounting for the slope of the bottom to the desired degree 
of approximation. This technique includes the slope, explic- 
itly, in the perturbation expansion. The solution for the 
waves on a plane sloping bottom has been worked out to the 
second order by several authors (93). 



75 



The analytical method is generally more consistent 
because it accounts for the bottom slope in the bottom 
boundary condition to the same order of approximation as is 
maintained in the free surface boundary conditions. Small 
bottom slope and amplitude are assumed in the perturbation 
analysis. Although more attractive from an analytical point 
of view, this method has the inherent difficulty of requiring 
tedious computations for each.. particular case. Also, the 
rigorous boundary value problem approach is not readily 
extended across the surf zone. 

The energy method consists of solving the wave problem 
for a horizontal bottom and then extending these results to 
a sloping bottom by means of energy flux considerations. 
Hence, for short distances, in the region outside the surf 
zone, it is assumed that waves on a sloping bottom can be 
considered the same as on a horizontal bottom. Then, adja- 
cent increments of distance are connected together by means 
of the energy flux conservation equation. This results in a 
prediction of the wave height at any location outside the 
surf zone. Inside the surf zone, it will be assumed that 
the wave height is governed by the local water depth. 

The energy method allows for the inclusion of energy 
dissipative effects, such as bottom friction, and the focus- 
ing or spreading of energy due to refraction by changes in 
bathymetry and currents. These effects are believed to be 



76 



more important tlian the deformation of the water particle 
motion due to the bottom slope as given in the analytical 
procedure (93). 

The validity of the energy method can be proven by 
comparison with the results of the exact analytical solution. 
To the first order of approximation in wave energy, for the 
case of small bottom slopes, identical results are obtained 
for the propagation, of energy. This confirms the use of the 
energy flux equation to connect the solution for various 
depths for small bottom slopes, at least to the first order 
in wave energy. 

In the present approach, the energy method describes 
the wave field to the first order in energy (second order in 
wave amplitude). The solution to the energy equation is 
substituted into wave momentum conservation equations to 
obtain the working equations needed for predicting longshore 
current and hence sand transport. 

4- . 2 Review of Mass, Momentum, and Energy Conservation 

P'eri'odic Wave Flow ' 

A starting point for the analysis is a review of the 

conservation equations of mass, momentum, and energy fluxes 

applicable to unsteady wave motion. The analysis is not 

concerned with the internal flow structure of the fluid; 

hence, the derivations can be simplified by integrating the 

conservation equations over depth. Conservation equations 



77 



which have already been developed by Phillips (94) will be 
used and are presented in a similar format to Phillips. 

The conservation equations will be applied to wave 
motion, but they are equally applicable to general turbulent 
motion. The unsteady velocity field of the wave motion can 
be expressed in the same manner as in the treatment of 
turbulent motion as the sum of its mean and fluctating 
parts - 

Resultant Mean Fluctuating 

u = U.(x,y,t) + ul (x,y,2,t) ,w(x,y,z,t) (4.1) 

where (1,2) refer to the horizontal coordinates (x,y), 
respectively, and z is the vertical coordinate. The tensor 
notation is used only for horizontal components of water 
particle motion. The mean current is assumed uniform over 
depth for simplicity. The pressure term can be stated 
similarly. These expressions can be substituted into the 
mass, momentum, and energy equations, and the mean and 
fluctuating contributions identified. 

The conservation equations are to be averaged over 
depth and time (one can consider averaging over a few wave 
periods). For the case of waves superposed on a mean cur- 
rent, all the wave motion would be identified with the 
fluctuating quantity which, when integrated over the total 
depth, can contain a mean contribution due to the waves. 
The time averaging of the equations for a general development, 



78 



being over a short interval compared to the total time, does 
not preclude long term unsteadiness in the mean motion. 

The utility of the conservation equations derived by 
Phillips is that the terms involving the mean and fluctuat- 
ing quantities have been separated. This facilitates the 
understanding of the effect of the unsteady wave motion on 
the total flow phenomenon. They are a particularly useful 
aid in gaining phy-s-ical . insight _ into the complicated mecha- 
nisms taking place in the surf zone. 

In addition to the conserved quantities of mass, momen- 
tum, and energy, a fourth conservation equation is available — 
conservation of phase (conservation of waves) (95). 

The conservation" of phase will be derived for simplic- 
ity in relation to simple harmonic motion. The expressions 
to be derived are also applicable to more general wave 
motion which can be formed by the superposition of individ- 
ual Fourier components. The expression for the water sur- 
face profile for harmonic motion is given by 




1 



= 1,2 



(4.2) 



where j Cx^) is the amplitude, and the quantity Ck^x^ - cot) 
is called the phase function X. The wave-number k^ and 
frequency to can be defined in terms of the phase function. 



such that 




■3-X 



C4 . 3 ) 



79 



An important property of the wave-number can be seen immedi- 
ately by vector identity 

^xk=VxVX=0 (4.5) 
so that the wave-number is irrotational (see for example 
(94)). From equations 4.3 and 4 . M- , the kinematical conser- 
vation equation for the wave-number can be written 

8k. „ 

1 + 4iiL. = 0 (4.6) 



3t 8x 
1 

For a single wave component, the conservation of phase says 
that the rate of increase of the number of waves in a fixed 
length is balanced by the net inward flux of waves per unit 
time. If (1) can be expressed as a function of k^ and pos- 
sibly X, from local arguments, (such as assuming a uniform 
simple harmonic wave train locally), and further allowing a 
mean current U, then, 

0) = a(k,x. ) + k.U. (4.7) 

a is the local wave frequency and is the apparent frequency 

to an observer moving with the current. Substituting into 

Equation 4,6, Whitham (95) shows 

3k. 3k. 3k. 

^ + c . + U. ^-^ = - G. (4.8) 



3t gj 3Xj : 3Xj 1 

where 

„ 3U. 

G. = + k. ^ (4.9) 

1 3x. 3 3x. 



80. 



and 

c . = If- (^.10) 

c . is the speed at which the energy or values of k- are 
propagated, commonly called the group velocity. The individ- 
ual crests propagate with the local phase velocity c = ^. 
The notation shall be adopted where, if c or k appear with- 
out vector notation, it is understood that they represent 
the modulus of the vector quantity. 

Equation 4.8 says that the rate of change of the wave- 
number k^, following a point moving with the combined group 
and convective velocity, is equal to - G^. Changes in 
are due to variations in the mean current and bottom con- 
figuration. These equations, which state the kinematical 
conservation for the wave-number, hold for any kind of wave 
motion (Phillips (94)). 

The general conservation of total mass per unit area 
can be expressed 

|^pD.|_S. .0 i = 1,2 (^.11) 

i 

D is the total averaged depth of water (which can include a 
mean elevation "n, above (or below) the still depth h) so that 

D(x,y,t) = (n + h) (4.12) 
The overbar is used to signify a time average. The total 
mass flux can be partitioned into its mean and fluctuat- 
ing components 



81 



M. = M. + M. (H.13) 
111 

The mass flux per unit width of the mean flow from Phillips 
(94) is 



M. = 
1 



_^ pU^ dz = pDU^ (1+.14) 



The mass transport of the fluctuating wave motion is 



M. 

1 



n 

pu! dz ■ (4.15) 

-h ^ 



The equation defining the conservation of horizontal 
momentum is derived by integrating the momentum equation, 
including shear stresses, over depth and averaging in time. 
The balance of total momentum per unit area can be expressed 
as per Phillips (94) 

—M. + ^ (U.M. + S. . ) = F^. + R. (4.16) 
3t 1 9Xj X : X3 px X 

Here M- denotes the total horizontal momentum per unit area. 

X 

Hence, the first term on the left represents the rate of 
change of the total mean momentum per unit area which in- 
cludes both the current momentum and wave momentum. 

The total mean transport velocity can be expressed in 
terms of the total mass flux per unit width as 

M. M. 

U. = = U. + (4.17) 

X pD X pD 

The second term on the left of Equation 4.16 expresses the 
momentum flux of a steady stream having the same mass flux 



82 



per unit width and mean .transport veJLojcity as the 
actual flow together with an excessive momentum flux term 
S^j arising from the superimposed wave motion, where 



^ij 



^ M.M. 

(pu!_u! + p6^j) dz - 1/2 PgD^fi^j - ■ (4.18) 



-h 



and is the Kronecker delta. This term is referred to as 

a "radiation stress" term by Longuet-Higgins and Stewart 
(91). 

The term F . is given by 

Fp. = - pg(n - h) II- (^.19) 

and represents the net horizontal force per unit area due to 
the slope of the free water surface. 

R. is the time mean averaged shear stress which must be 

X 

included in any realistic treatment of the surf zone where 
dissipative effects occur. Considering the shear stresses 
on a column of water (see Figure 4.1) and integrating over 
the depth, R^^ can be expressed 



R. = 



n n 
dz + 



dz i,j = 1,2 (4.20) 

d Z 



where t.. includes the combined stresses of waves and cur- 
rents. The shear stress on the horizontal plane can be 
integrated over depth, such that 



Figure 4.1. Shear Stresses Acting on the 
Faces of a Water Column 



84 




85 



3t 



dz 



T .- T, . 



(4.21) 



where the subscripts refer to surface and bottom. 

The Leibnitz rule of integration must be employed to 
integrate over depth the stresses occurring on the vertical 
faces of the water column. Evaluating the shear stress on 
the vertical faces and taking the time average of the terms 



3t 



3 X 



il d: 



8x, 



an 9h 

^ji^^ " "^jin ~ ^jih 3x. 



-h -h 

Thus, the general shear stress term is given by 



(4.22) 



R 



3x 



T . . dz - T . . 

31 



3n 



- T 



3h 



. . ^ . . ^ + T . T, . (4.23) 

3in 3x^ jxh 3x^ ni — hi 



-h 



3 " 3 

i,3 = 1,2 

The equation for the conservation of total energy can be 
partitioned into the energy contributions from the mean and 
fluctuating parts with the aid of the conservation of mass 
and momentum equations. In the following development, it is 
convenient to work with the energy balance' 'for the' fluctu- 
ating motion alone, which can be stated as per Phillips 
(94) . 



86 



■ Mf . U-M? 3U. 

In separating the mean and fluctuating contributions , the 
mean energy density has been represented as the total energy 
density of an "equivalent" uniform flow with the same depth 
and mass flux as the actual flow, plus the energy density of 
the fluctuating motion, minus an additional term represent- 
ing the difference between -i^ie- BTiergy density of the actual 
mean motion and that of the equivalent uniform flow. Then, 
the first term of Equation 4.24 is the rate of change of the 
energy density of the fluctuating motion minus the correc- 
tion term. The second term represents the convection of the 
fluctuating motion F, minus the correction term. The trans- 
port of energy by the fluctuating motion is given by 



F. = P 



n 

2 



ul_ (l/2(u!^ + w'^) + g(z - n) + p) dz (4.25) 



-h 

which represents the rate of work done by fluctuating water 
particle motion (for turbulence, this would be the work done 
by the Reynolds stresses) throughout the interior region of 
the flow, plus the work done by the pressure and gravity 
forces. The last term on the left of Equation 4.24 repre- 
sents the rate of work of the fluctuating motion against the 
mean rate of shear. Dissipative effects of turbulence have 



87 



also been allowed where £ jLs the ra-te of energy dissipation 
per unit area. 

The energy budget for the mean motion is given by 

1^ (1/2 U.M. + 1/2 pgCn - h^)) + 

3S 

3I- M. (1/2 G.2 . gW) . U 3^ = U.T^. . (4.26) 

The terms of this equation represent, respectively, the rate 
of change of the kinetic and potential energy of the mean 
motion, the transport of the total mean energy, and the rate 
of work by mean motion on the fluctuating motion and bottom 
shear stresses. 

In the development of the conservation equations, no 
restrictions were plac-ed on the wave slopes or amplitudes. 
Also, no restrictions were placed on the fluctuating motion 
so that the equations are equally applicable to wave or 
turbulent motion. 

4 . 3 Review of Wave Field Equatiohs 
The descriptive equations for the wave field are de- 
rived by solving the linearized boundary value problem over 
a horizontal bed. This solution can then be extended to a 
higher order by perturbation techniques. The development 
presented here will retain terms to the second order in 
amplitude (first order in energy and momentum) and neglect 
all higher order terms. The wave solution can then be 
substituted directly into the conservation equations 



88 



providing a means .for describing the wave-induced mean 
motions. In making this substitution and dropping all terms 
of orders higher that the second, only knowledge of the 
first order (linear theory) wave water particle velocities 
and surface elevation is necessary. This is because, in 
expanding and then averaging over the period, the terms 
involving higher order quantities in velocity and surface 
elevation- go to zero. The pressure must be known to the 
second order in wave height; however, the average second 
order pressure component can be determined from the first 
order water particle velocities and surface elevation terms. 
Thus, only the linear wave solution small amplitude is 
required. 

In utilizing linear theory, it is assumed that the 
motion is irrotational and that the fluid is incompressible 
and inviscid (see for example Kinsman (96)). These have 
been shown to be good assumptions (96). The fact that the 
linear theory is a good approximation is demonstrated by its 
success in describing many observed phenomena (96). 

Assuming simple harmonic motion, the surface elevation 
is (95) 

n = Y (k^x^ - CTt) i = 1,2 

where the wave-number components are 
k = k cosa 

X (4.27) 

k = k sina 



89 



The arbitrary angle of wave incidence a is measured between 
a line parallel to the contours and the wave crests. The 
velocity potential for the first order solution is given by 
(96) 

H cosh k(h + z) ^.^ _ (^_28) 

2 ck cosh kh x x 

The velocity is related to the gradient of the velocity 
potential 

= - (4.29) 

H g i cosh k(h + z) ^^^/^ ^ qn^ 

U . = TT ^ — r — r-r cosCk.x. - at J (.M-.ou; 

X 2 c k cosh kh x x 

The frequency equation relates the frequency to the local 

wave-number and water depth (96) 

= gk tanh kh (4.31) 
The pressure can be determined as a function of the depth to 
the second order by integrating Euler ' s vertical momentum 
equation over the depth and retaining terms of second order. 
Only knowledge of the first order water particle velocities 
and surface elevation is required to determine the mean 
second order pressure term. The time-mean pressure is given 
as 

2 

— 1 2 tt2 sinh k(z + h) /,. 

p = - pgz - g-pa H 2 (4.32) 

sinh kh 

where the first term is the hydrostatic contribution, and 
the second term is the second order mean dynamical pressure 
component . 



90 



The group velocity .-of- ±±ie waves can be determined from 
Equation 4 . 9 

(c ). = ^ = £ n ^ = c,n (4.33) 

g 1 k dk k k 1 

where the transmission coefficient is 

n = 1 (1 + .^^^ ) (4.34) 

" 2 sinh 2kh^ 

For deep water (infinite depth), n = and, in shallow 
water, n approaches 1. 

The energy density of the waves is proportional to the 
wave height squared and is equally partitioned between the 
potential and kinetic energy, such that the total energy 
density is 

E = ipgH^ (4.35) 

Recalling that the group velocity expresses the speed of 
energy propagation, the energy flux of the wave motion is 

F. = E c . (4.36) 

Assuming negligible wave reflection and a mildly sloping 
bottom, Equations '.4.3 3 and .4. 3 5 "' can be substituted 
directly into Equation :4.35: to obtain the flux of wave 
energy. 

The mass transport of the waves can be determined by 
recalling Equation '4.15. in which the fluctuating velocity 
was integrated from the bottom to the surface. Since the 
water surface elevation is unknown, the integrand is ex- 
panded in a Taylor series about the mean water surface level 

n 



91 



M. = 



pul^ dz + 



-h 



3ul (n) 

p(u!(n) + z -rr-^ + ...) dz (4.37) 

1 o Z 



n 

where the time mean average of the mass transport below the 
mean surface is zero (first integral), and only terms of 
second order in amplitude will be retained. Substituting 
for the wave profile and water particle velocity of the wave 
motion, Equations 4.2, and ■:4.30:, the ftiass transport of 
the waves is 

TII = 5. _i (4.38 ) 



M . = 0 nu . — - — T- 

1 2-r\ c k 

Simple harmonic motion has been selected to describe the 
waves because experience has shown that this solution gives 
fairly good results for the deep water case. Inside the 
surf zone, however, the approximation is not good, but the 
assumption of simple harmonic wave trains will be retained 
with the exception of certain modifications on the celerity 
and amplitude. The wave field could be specified using the 
cnoidal or other higher order wave theories, but the accu- 
racy gained would not justify the increased complexity of 
the resulting equations (93). 

To simplify the working equations a determination of 
mass transport and energy is first arrived at. The schema- 
tic of the coordinate system used is shown in Figure 4.2. 



32 



ELEVATION VIEW 




Figure 4.2. Schematic of the Surf Zone 



93 



Although a longshore current distribution is shown, no prior 
knowledge of its actual shape has been assumed at this time. 

Due to the absence of any y-dependence , the mass con- 
servation Equation (M-.H; reduces to 

= 0 (4.39) 

3x 

Integration gives 

M = constant =0 (4.40) 

X 

The constant of integration is equal to zero since the beach 
forms a boundary in the x-direction. This then says using 
equation 4.17 that 

U = - = - cosa (4.41) 

pD pcD 

which states that there is a mean reverse current balancing 
the mass transport onshore due to the wave motion. This must 
be true everywhere, both inside and outside the surf zone, 
to ensure that there is no accumulation of mass or growth of 
currents in the y-direction in order to maintain steady- 
state conditions in accordance with the original assumptions. 

The determination of the energy distribution is neces- 
sary as a means of relating the wave heights at various 
depths. The energy, in turn, must be related to the local 
angle of wave incidence. The angle of wave incidence is 
affected by refraction which can occur since the waves are 
allowed to approach at an arbitrary angle of incidence to 
the bottom contours. The problem is further complicated 



9^ 



.since a shear flow is allowed which can also produce wave 
refraction. In the present model this shear flow effect on 
wave refraction will be assumed to be a minor effect and 
hence neglected. This is comparable to assuming that the 
longshore directed current is small in comparison to the 
maximum wave orbital velocity. 

The general statement of the phase relation gives an 
expression r.elating. thje., frequency, mean current motion, and 
wave angle. Equation '. M-.?' can be expanded to give 

(D = cr + Vk sina + Uk cosa (h.^■2) 

The wave-number was shown to be irrotational 
3k Sk 

^ - ^ = 0 (4.43) 
3x 3y 

Since the wave length and amplitude of the waves are indepen- 
dent of the y-direction, the gradient of the local wave- 
number in the y-direction is zero. 
3k 3k 

Integrating the gradient of k^ in the x-direction gives 

k sina = constant = k^sina^ (4.45) 

o o 

where the subscript "o" refers to conditions in deep water. 
This is a statement that, for straight and parallel contours, 
the projection of the wave-number on the beach is a constant. 
Assuming zero current in deep water, a general expression 
for the celerity of the waves can be derived where, as 
before ; 



95 



0) = + V sina + U casa) (4.46) 

and utilizing Equation '!4.45? 

c = (—JS. V) sina - U cosa (4.47) 

sma 

o 

For U = V = 0, this expression simplifies to Snell's law for 
wave refraction. 

The conservation of energy for the fluctuating motion, 
Equation ;4.24'', can b-e expanded for steady-state conditions 
to give 

. U M ^ U M ^ 

f-(EU + Ec - -4-4 ^-4r-) + S 1^ + S ^ = - z (4.48) 

3x X gx 2pD 2pD xy 3x xx 3x 

where the gradients in the y-direction are zero. This 
equation states that the change in energy flux, due to 
currents and waves plus the work done by the excess momentum 
flux on the straining motion, is equal to the energy dissi- 
pated by turbulence and work done on the bottom. The prod- 
uct of a stress times rate of strain is a quantity that can 
be associated with power per unit volume. The last two 
terms on the left of the energy equation can be interpreted 
in this context where the excess momentum flux tensor then 
represents a stress and the velocity-shear a rate of strain. 
Longuet-Higgins and Stewart (91) named the excess momentum 
flux tensor a " radiation stress . " 

The excess momentiim flux tensor can be determined by 
substituting the wave expressions into Equation '4.18' . In 



96 



terms of energy, group velocity, and wave speed, an expres- 
sion applicable to both inside and outside the surf ^one is 
given by 



c 2c 

E-^os^a + ^i—^ - 1) 
c 2 c 



5. — S. sin2a 
2 c 



1 -S. sin2a 

2 c 



c c 

E sin^a + |(2-^ - 1) 
c 2 c 



(4.1+9) 

Effects of turbulence have not been included. 

Referring to Equation (4.41) and (4.49), it can be seen 

that U and S are of order E. The first and last terms of 

Equation (4.48) involving the product of these terms are of 

order (E ) and, hence, will be ignored in this analysis. 

From the result of the conservation of mass, Equation (4.41), 

U = 0, so that the third and fourth terms of the energy 
X ' 

equation are zero. Substituting for S and F , and retain- 
ing only terms of first order in energy, the energy equation 
reduces to 

(4.50) 



8 N J. -n sin2a 3V _ 
-r— (Ecn cosa) + En — ^ ir— - - e 

dX / dX 



where the substitution, c = cn, has been made. Substituting 
for the wave celerity, as given by Equation (4.47), yields 



(— - V) sina cosa - U cos a 

sma 

o 



3V 



+ En cosa sina = - £ (4.51) 

d X 



97 



Again recalling that U is .of. ordBr E, and that all higher 
order terms involving the product of U can be neglected in 
this analysis, the energy equation can be written 

°_ En( . ° - V) sina cosa + En cosa sina = - £ (4.52) 
3x sm 

o 

Expanding and cancelling terms, gives 

V) I- En sina cosa = - £ (4.53) 

sma 8x 
o 

Far outside the surf zone, energy dissipation due to bottom 
effects can be ignored, and the energy losses due to turbu- 
lence can be assumed negligible. Since the term in the 
brackets of Equation ..;4.5'3' is nonzero, the result of the 
energy equation outside the surf zone, assuming no energy 
dissipation, is 

En sin2a = constant (1.54) 
The relative amplification of the wave energy is then given 
by 

_ n sin2a 

^ = ° . ^ • (4.55) 

E n sin2a 
o 

Since the term involving U in the celerity equation resulted 
in the product of higher order terms that are not included. 
Equation '4.47' can be written with the additional assump- 
tion that the velocity V is minor outside the surf zone, 
c 

c = (~ — ) sina (4.56) 
sma 

o 

or in terms of the wave angle 



98 



c 



(4.57) 



sxna = 



sma 



c 



o 



o 



which is a statement of Snell's Law of Refraction (96). 
Equations (4.53; and '4.55:'give a complete description of 
the wave amplitude and direction outside the surf zone. 

Equation :4.4-7- can be combined with Equation '■4.55: to 

give 



This states that the onshore component of energy flux is 
constant outside the breaking zone. The changes in energy 
density E, or wave height, can be determined from this 
equation as a function of the local wave-number and water 
depth. For decreasing depth as the waves approach the 
shore, the local wave length and the angle of incidence 
decrease. The effect of shoaling is determined by the group 
velocity. The group velocity initially increases slightly 
so the energy density decreases; the group velocity then 
decreases resulting in a continual increase in the energy 
density towards shore. A maximum wave height occurs at 
breaking. Due to the change in wave angle, which is the 
result of refraction, the wave crests become more nearly 
parallel to the beach. The energy density is less for waves 
approaching at an angle to a constant sloping beach than for 
waves whose orthogonals are normal to the beach because wave 
refraction results in divergence of wave energy. 



E c cosa 
g 



c 

- E — cosa •= const 



o 2 o 



(4.58) 



99 



As has been shown, there is an excess of momentum 
flux due to the presence of the waves. For conservation 
of momentum flux, there must be a force exerted in the 
opposite direction such as a hydrostatic pressure force 
or bottom shear stress to balance this excess momentum 
flux. It has been shown that outside the surf zone the 
component of excess momentum flux directed perpendicular 
to the contours is balanced by mean water level set-down, 
and inside the surf zone that such a balance leads to a 
wave set up (97). 

These effects have shown (9 7) to have a minor 
effect (second order) on water depth outside the surf zone 
and for all practical purposes can be neglected in that 
region. 

The seaward edge of the surf zone is usually de- 
lineated by the point where the waves first start to break 
which is reached when the particle velocity at the wave 
crest is equal to the wave celerity. Inside the surf zone, 
the waves are unstable, and the fluid motion tends to lose 
its ordered character. Waves break in different ways, 
depending primarily upon the wave steepness and slope of 
the beach. The manner in which they break has a very 
definite influence on the hydrodynamics inside the surf 
zone which, in turn, affects such quantities as the sedi- 
ment transport, longshore currents, and wave runup. 



100 



Based on observations by Galvin (76), breaking waves 
are usually classified as spilling, plunging, or surging. 
Spilling occurs when the wave crests become unstable, curl 
over slightly at the top, creating a foamy advancing face. 
Plunging occurs on steeper beaches when the wave becomes 
very asymmetric; the crest curls over, falling forward of 
the face resulting in the creation of considerable turbu- 
lence, after which a bore-like. wave. front develops. Surging 
occurs when the wave crest remains unbroken while the base 
of the front face of the wave, with minor turbulence gen- 
erated, advances up the beach. 

There is a continuous gradation in the type of break- 
ing, and Galvin (76) found it convenient to add a fourth 
category of collapsing to describe a type intermediate to 
plunging and surging. He performed extensive laboratory 
investigations to quantitatively classify breaking waves 
according to the wave and beach characteristics . Combining 
his results with earlier works, he grouped the breaker type 
depending on beach slope tanB, wave period T, and either 
deep water or breaker height H. The breaker type can be 
established as shown by Battjes (77), by the dimensionless 

(H /L )-^2 (H^/L^)^^^ 

variable, — 2 — 2 or , . As either of these 

tanp tanp 

parameters increases, breaker type changes from surging to 

collapsing to plunging to spilling. Spilling breakers are 

associated with steep, relatively short period waves and 



101 



flat beaches; plunging breakers ..are .a^-sociated with waves 
of intermediate steepness and the steeper beaches; and 
surging breakers are associated with waves of small height 
and steep beaches. 

On natural beaches, breakers classed as spilling are 
most commonly observed, followed in decreasing order of 
frequency by plunging, collapsing, and surging. In the 
laboratory tes-ts, spilling ..bjr>eakers are relatively rare 
compared with collapsing and surging breakers because 
slopes used in laboratory tests are usually steeper than 
slopes commonly found in nature due to the physical limita- 
tions of space. 

The breaking index curve provides a relationship be- 
tween the breaking depth , the breaking wave height , 
and the wave period T. In shallow water, the relationship 
simplifies to H'^^/D^ = < , a constant. Longuet-Higgins (M-6) 
compiled breaking wave data from several sources (9 8-106) 
including lab and field data. He found a fairly good 
correlation for breaking waves as compared to the breaking 
wave criteria predicted by the solitary wave theory. The 
solitary wave theory predicts a value for k of 0.78, while 
other theories predict slightly different values. Theo- 
retical values range from 0.73, found by Laitone (105), fo 
cnoidal wave theory, to a value of 1.0, found by Dean (10 5 
using a numerical stream function theory. Experiments on 
steeper laboratory beaches show that the value of k can be 



102 



much larger. All th.e theor-etical values have been calcu- 
lated, assuming a flat bottom, and correspond to beaches 
with very gentle slopes. A summary of various investiga- 
tor's findings zis presented in Table 4.1. The important 
point is that the breaker height is governed by the depth of 
water . 

The energy loss due to the wave breaking process is 
dissipated in the generation .ol turbulence and heat, bottom 
friction, dissipation, percolation, and viscosity. The 
waves in the surf zone constitute a non-conservative system 
in which the use of potential flow theory is no longer 
valid. In fact, there is no analytical description avail- 
able for the waves in the surf zone. Gross assumptions are 
required. The linear wave theory will be retained as the 
input to the conservation equations, but with modification 
to the wave amplitude. . The wave height inside the surf zone 
as noted is controlled by the depth of water and is of the 
same order of magnitude. 

Spilling breakers lend themselves to a physical treat- 
ment since the potential energy and momentum flux of the 

« 

waves inside the surf zone can be expressed approximately in 

analytical form. If the beach slope is very gentle, the 

spilling breakers lose energy gradually, and the height of 

the breaking wave approximately follows the breaking index 
given previously. The height of the wave is then a function 
of the depth. 



103 



H, /2 

Table "+.1. Observed and Theoreirical "Values of j = . 



Investigator 


tanB 


K 

2 


IC 

2" 




Observed Values 


Average 


Putnam et al. (44) 


0.065 
0.098 


0.37 
0.36 






0.139 
0.143 


0.32 
0.37 


0.35 




0.144 


0.32 






0.241 


0.35 






U . Z D VJ 






Iverson (98) 


0.020 


0.41 






0.033 
0.050 


0.38 
0.42 


0.44 




0.100 


0.52 




Larras (99) 


0.010 


0.34 






0.020 

0.091 


0.37 
0.43 


0.39 


Ippen and Kulin (100) 


0.023 


0.60 


0.60 


Eagleson (101) 


0.067 


0.56 


0 . 56 


Galvin and Eagleson ( 54 ) 


0,104 


0.59 


0.59 


Bowen (47) 


0.082 


0.45-0.62 


0 . 56 


Values Determined 


from Solitary Wave Theory 




McCowan (102) 


0.000 


0.39 




Davies (103) 


0.000 


0.41 




Long (104) 


0.000 


0.406 




Values Determined by Cnoidal 


Wave Theory 




Laitone (105) 


0.000 


0.37 




Values Determined by Numerical Wave Theory 




Dean (106) 


0.000 


0.50 





104 



In the present model, it will be assumed that the waves 
act as spilling breakers inside the surf zone and that they 
follow the breaking index, k = 0.80. The wave height inside 
the surf zone is then given by 

H = kD (4.59) 

It is further assumed that the kinetic and potential 
energy are equally partitioned so that the total wave energy 
can be described in .texims of the wave height which is a 
function of the depth 

E = I pgK^D^ (4.60) 

This is a non-conservative statement of the energy distribu- 
tion within the surf zone. 

The waves inside the surf zone are assumed to retain 
their simple harmonic character so that the wave profile 
and water particle velocity are described by 

n = Y cos (k^X^ - (jOt) 

(4.61) 

u = 75- ^ r-^ cos (k.x. -(jot) 
2 c k 1 i 

The expression for the horizontal water particle velocity is 
based on the Airy wave theory and has been simplified for 
shallow water. 

In very shallow water, the waves are non-dispersive 
with the wave speed being only a function of the depth. In 
keeping with the linear theory approach to calculation of 
wave energy it will be assumed that wave celerity is ade- 
quately described by linear wave theory 



105 



c = (gD)-^^^ (4.62) 
It has been found experimentally in shallow water 
breaking, wave celerity can be better approximated by soli- 
tary wave theory (102); 

c = (g(H + D))^''^ = (g(l + <)D)^^^ (4.63) 
It is felt though, that since wave height is linear in depth 
any way and therefore there is only a difference of a con- 
stant between the two descriptions of celerity, the linear 
wave model should be used for consistency of theory. 

M- . M- Proposed Sand Transport Model 

4.4.1, Longshore Current Estimation 

The derivation of the longshore current to use in the 
sand transport equation parallels an approach similar to 
Longuet-Higgins (46), and Thornton (30), in that a linear- 
ized bottom stress in longshore velocity is found from which 
longshore current can be postulated. The present approach 
is different only in that a wave orbital velocity shear 
stress friction factor is used in the bottom shear stress 
term, and wave set-up is included. 

In order to formulate the longshore current problem 
some assumptions concerning the flow conditions are restated. 
In the present development as in the simplified models of 
Longuet-Higgins (46) and Bowen C47) it is assumed that 
(1) longshore currents are steady, no time dependence, 



108 



(2) homogeneous and incompressible fluid; (3) hydrostatic 
pressure -.CLO 7>, l4)rCD'rioli-S force is negligible; and (5) currents 
are sufficiently small such that wave current interaction is 
negligible. 

Using the assumption of uniform motion in the y direc- 
tion and no time dependence. Equation 4.16 can be reduced to 
the following forms 



0 = - pg (n+h)4^ - T, + T 

' 8x hx rx 



(4.64) 



n 
-h 

where 

as 



T . - T-, + T 

xydz hy ry 



"rx 3x 
SS 



(4.65) 



T 



'ry 3x 
and 

D = n + h (4. 65) 

The quantity under the integral sign can be posed 
similar ily to a turbulent Reynolds stress as 



T dz = y D 1^ (4.67) 
xy ex dx 

-h 

Then assuming that the onshore time average bottom 
stress is zero, equation 4.6 4 can be reduced to 



107 



0 = -pg(n+.h'>-|^-+ T 



rx 

(4.68) 

8x ex dx hy ry 

From equation 4.4 9 with the shallow water assumption it can 
be found that 

S^^ = I (3 - 2sin^a) (4.69) 

and 

c 

S = E cosa sina (4.70) 
xy c 

Also, a relationship for Snell ' s Law of Refraction (i.e., 
Kinsman (96)) is of the form 

sina 

sma o 

= constant = (4.71) 

c c 

o 

which can be included in the above term to obtain 

sina 

S = F (4.72) 

xy X c , 
o 

where F is the onshore directed component of wave energy 
flux 

F = E c cos a (4.73) 
X g 

and was shown to be constant outside the wave breaking zone. 

Therefore, the quantity S is also constant outside the 

xy 

wave breaking zone and eaual to the deep water value S , 

xy 

where 

^xy ~ ~2" ^°^^o ^-^•'^'^o (4.74) 



108 




Figure 4.3. Schematic Plan and Section of the Nearshore Region 



109 



At the shoreline (where D=0) since wave height H is 
proportional to depth D the above quantities become 
F = 0 

X 

S =0 
xy 

and the total external lateral force on the breaking zone 
equals 

E 



Total lateral forc^. on .breaking zone = — sin2a^ 



(4.75) 



Considering now the balance of momentum in the onshore 
direction of the water between the breaker line and the 
shoreline as expressed by Equation 4.6 8 with the expression 
given previously 

H = kD = ic(n + h) (4.76) 
one can derive the following equation 



dx 



2(3-2 sin a, ) , 

K b d 



16(n + h) 



dx 



(n + h) 



(4.77) 



or 



= _ K = - K tang . 
dx dx 



where 



K 



2 2 
K (3 - 2 sina, ) 
b 

8 + ic'^O - 2 sin'^a, ) 

b 



(4.78) 



(4.79 ) 



If a new horizontal .-cGordinate , x, is introduced, with 
its origin at the point of maximum set-up x = ~^s' "then 

X = X + X , and rf + h = (1 - K) tan 6 x. 

s 

Assuming "=0 at the breaker line, one can integrate 
equation 4.77 to find n" and the total depth D across the 



110 



surf zone. In reality, it has .been.-shown that there is a 
setdown at the breaker zone (i.e., rf = negative value) but 
the set down has been shown to be negligible for all prac- 
tical purposes (97). 

Inside the breaker zone F gradually diminishes toward 
the shoreline. A consideration of the momentum balance 
between two planes x = x-^^ and x^ + dx parallel to the shore- 
line and - separat-ed by a distance dx shows that the net 
stress T^^ per unit area exerted by the waves on the water 
in the surf zone is given by 

^ry = - ^S^y/^^ ^^'SO^ 
and using equation 4.72 this equation becomes 

8F sina 

O 

3^x 

where g— ^ denotes the local rate of energy dissipation in 

the onshore direction. This equation notes that the local 

stress exerted by the waves is directly proportional to the 

local rate of dissipation of onshore directed wave energy. 

Outside the breaker zone the mean stress vanishes as S is 

xy 

constant as shown previously. 

In some situations the loss of wave energy can be 
attributed to bottom friction (due mainly to the orbital 
velocity of the waves). Observations by Munk (108) that in 
the surf zone the breaker height is proportional to the mean 
depth suggests that under normal circumstances most of the 



Ill 



loss of wave energy is.due.io wave breaking though not to 
bottom friction. 

It is important to remember that the large portion of 
the wave energy expended in the surf zone is not used in 
moving sand in the longshore direction but is dissipated 
mostly through the turbulence mechanism. Other investi- 
gators have correlated the entire energy flux dissipated 
throughout the- ^surf ^one- ■■with the amount of sand transport 
moved as suspended and bed load (30), while only a small 
fraction of this energy is readily available for actual sand 
transport in the longshore direction. 

It was shown previously that 
H = kD 

where k is a constant between 0.7 and 1.0 (see Table 4.1). 
Using linear wave theory where kD<<l we have 
c = /gD - c 
and therefore we have 

T- tH^ 3/2 ^5/2 ,„ 

F = ~— c cosa = -5— pg D cosa (4.82) 
X 8 g 8 '^'^ 

and from equation 4.81 
3 S 

rn xy 5 2 / T^^3/2 dD sina cosa 

T = - ^ ^ = - K p ( gD ) -3 

ry 3x 16 * dx c 

= - <^yD 4^ sina cosa (4.83) 
16 ' dx 

where ^ = ^ = (1-K) tan3 is the modified local bottom slope, 
dx 



112 



Using equations H.82 and the linear shallow-water 
theory, we can also express equation 4.83 in terms of the 
maximum horizontal bottom orbital velocity given by 

^hwm = f '^/^^ = f^"/^^ = f ^ (4.84) 
Then we have simply 

5 2 

= - IT P^hwm ^^^^ sina cosa (4.85) 

Seaward of the .breaker line the onshore component of 
energy flux is constant (i.e., no dissipation assumed in 
the present case) and thus 

T =0 
ry 

The tangential stress excited by the water on the 
bottom will be assumed to be given adequately by a relation 
of the form 

= f f |u^ |u^ (4.86) 

Where u^^ is the instantaneous velocity vector near the 
bottom and f ' is a friction factor. 

If there were no longshore velocity, and if the ampli- 
tude of the motion were small and the bottom impermeable, 
the horizontal orbital velocity would be expected to be to- 
and-fro in the same straight line, making an angle with the 
normal to the shoreline (see Figure 2a in Longuet-Higgins 
(46) ) . 

The frictional stress would then be given by 

"^h = f l^hw l^hw (4-87) 



113 



which would then vanish ±n ■ the, mean - BJZcoTdlng to linear wave 

theory. Here f , has been assumed for f since the motion is 

w 

entirely due to waves . 

If a small component of velocity in the longshore 
direction V is added to the orbital velocity, the frictional 
stress no longer vanishes in the mean. Assuming a is small, 
the component of velocity V is almost perpendicular to the 
orbital velociiiy. To first urdeT, the magnitude of the 
velocity 

l^hl = ^l^hwi' ' \^\^^^^^ (4.88) 
is unchanged but the direction of the bottom stress is 
changed by a small angle - V/|u^^|. This leads to an addi- 
tional stress in the y direction given by 

V = J l"hwl l%„l 1:^1 ' ("-sa) 

Physically, when the orbital velocity is onshore, the 
direction of the bottom stress is inclined more tovzard the 
positive y direction (when V is positive); when the orbital 
velocity is offshore, the bottom stress, now almost in the 
opposite directions, is again more toward the positive y 
direction. This is shown in Figure 4.4. 

Taking the time mean value of the longshore shear 
stress the following relationship is found: 



114 



V (4.90) 



hy 2 

as is assumed sinusoidal as per linear wave theory 



TT ^hwm (4.91) 



As the friction coefficient was determined in accordance 

with a small V comparatively to the orbital wave velocity 

u^^, the bottom shear is dominated by the wave ortibal shear 

stress and the proper friction coefficient is assumed to be 

f ^ (i.e., the bottom shear stress is governed by the wave 
w ^ ^ 

orbital motion being the dominant shear stress term) . 

The friction factor f has been found (65) as a func- 

w 

tion of the bottom wave particle amplitude Reynolds 
number and the roughness to wave particle amplitude ratio. 

The equation of motion in the longshore direction can 
be written as before 

8 . ^ 9V. 



0 = T , + ^ (y B--^) - T, (4.92) 
ry 9x ex 3x hy 



where in the surf zone T^^ and x^^ are given by equations 
4.85 and 4.90, respectively. 

Little is known as to the lateral "mixing" coefficient 
U which spreads momentum laterally across the surf zone. 
A few theories exist, but are based on very limited data 
(109). As the spread of lateral momentum across the surf 
zone only changes the lateral distribution of the longshore 
velocity and its maximum value, and the required final 



115 




Figure 4.4. 



Shear Forces Due to Oscillatory Wave 
Velocity and Longshore Current 



116 



velocity distribution is to ha J.ntagi?ated -across the surf 
zone, it will first be assumed that y is equal to zero. 
This is equivalent to saying that there is no shear coupling 
between adjacent water columns taken across the surf zone. 
This may be contrary to what various experiments on long- 
shore currents have shown (109) but will be sufficient for 
initial development of the intended longshore current equa- 
tion. It will l-airer be -shown that inclusion of an u 
value not equal to zero and proper accountability for the 
longshore current distribution across the surf zone only 
changes the integrated sand transport equation by a constant, 

Another way of neglecting the lateral eddy viscosity in 
the longshore current equation is to suppose that the ex- 
change of momentum by turbulence is negligible in comparison 
with that due to waves (M-S). Then, in general the second 
term on the right of equation 14, 92 can be neglected in 
comparison with the first. 

There remains a balance between the first and third 
terms : 



T7~ = T (4.93) 
hy ry 

Substituting from equations 4.8 5 and 4.90, we have in the 
breaker zone 

-f£-u^ V = -^pu, ^ (1-K) tanS sina cosa (4.94) 
TT w 2 hwm 4 nwm 

and 



117 



V = ^-J- u, (1-K) tan-B- sin-a 'cosa (4.95) 

4-1 nwm 
w 

Using the linear theory relationship between and wave 

celerity c the equation for longshore current velocity 
becomes 

V = ~ ^ gD(l-K) tang (iiBE) cosa (4.96) 

w 

where c = /gD as before. By Snell's law the next to last 
factor in the above equation is constant. Thus, in this 
derivation, the longshore velocity is simply proportional to 
local total depth D. 

If it is assumed that the shallow water theory is valid 
as far out as the breaker line where the depth D is equal to 
Dg , the longshore current, in the absence of horizontal 
mixing, can be written as 



° < °B 



0 ^ ^ 



(4.97) 



where 

Vg = velocity at breaking line 

= 1^ ^ /gU^ (1-K) tang sina^ cosa^ (4.98) 
w 

The velocity distribution represented by equations 4.9 7 
and 4.98 is a triangular distribution with a maximum veloc- 
ity at the breaker line. As noted earlier the turbulent 
mixing as represented by a "mixing" eddy cofficient y has 
been arbitrarily assumed as zero which provides a 



118 



discontinuity in the velocity. distri±i.utioji .profile at the 
breaker zone. 

Both Bowen (47) and Longuet-Higgins (1+6) have solved 
for the case of lateral shear mixing across the surf zone by 
assuming an eddy viscosity based on mixing length theory. 
The present extended model exactly parallels the solution 
given by Longuet-Higgins (46), the only difference being 
that wav^ set up is account-ed for. In Longuet-Higgins 
solution (46) and the present solution y is assumed to be 
given by a form of mixing length times mixing velocity 

= Npx /gD (4.99) 

where N is a dimensionless constant; x is a mixing length, 
increasing linearily from the shoreline boundary similarly 
to the Prandtl hypothesis; and, the mixing velocity is 
proportional to the horizontal water particle velocity which 
is proportional to wave celerity = /gjD. The solution of 
equation 4.9 2 under the consideration of lateral shear 
mixing is given in (46) and the solutions are shown here 
with the modification that wave set-up is included. The 
equations for longshore current distribution with wave set- 
up are 

V„ (B.X ^1 + AX) 0< X<1 
V = { ^ ^ p (4.100) 
Vg (B2X^2) 1< X < <» 

and by matching solutions at the breaker line 



119 



x=l 



V 



x=l 



8x 



x=l 



3 x = l 
3x 



(4.101) 



the following values are found 

1/2 

. + L> 

4 '16 p 



= . 3 , (4 , 1, 



„ _ 3 , 9 , 1.1/2 



B- 



B, 



A 



. A 



and where 



P2-I 
P1-P2 

Pl-1 
P1-P2 

2_ 

2-Sp 



2ttN(1-K) tanB 



KT 



(4.102) 
(4.103) 

(4.104) 

(4.105 ) 
(4,106 ) 

(4.107) 



w 



X = x/x, a dimensionles s distance 

x^ = distance to the breaker line from the shoreline 

As can be seen in the above equations at the value of p = 2/5 
a singularity exists in the above solutions. The solution at 
p = 2/5 is of a separate form (46), and is given by 

0<X<1 

(4.108) 

l<X<oo 



V = 



Vg (10/49X - 5/7X In X) 



Vg (10/49X - 5/2) 



As before, Vg equals the longshore current velocity at the 
breaker zone in the absence of lateral shear mixing and is 
defined in equation 4.98. 



120 



Experimental evidence in laboratory longshore current 
experiments (46) suggests a coefficient of p between 0.1 and 
0.4 when comparing laboratory longshore current measurements 
to theoretical values. 

4.4.2 Integrated Sand Transport Estimation 

The proposed sand transport model is an extension of an 
existing model of sand transport based on an energy-rtype- 
approach due to Bagnold (45) and postulated for unidirec- 
tional current flow. The present model extends Bagnold 's 
basic concept to the realm of sand transport along beaches 
parallel to the shoreline due to wave action. The concept 
can be extended to the case of sand transport along beaches 
due to other driving forces such as wind shear on the surf 
zone or shear due to tidal induced currents (i.e., hydraulic 
tidal currents in the vicinity of inlets). In the presently 
developed model though ' only the sand transport due to wave 
motion will be considered. 

The presently developed model also differentiates 
between bed load and suspended load sand transport and each 
component can be obtained independently, a distinct advan- 
tage not realized in many sand transport models (i.e., 
Einstein (110)) where the suspended load component is 
dependent upon the calculated bed load component. 

Now, defining the dry mass of a unit area of material 
to be m and the solid density of the material to be p , the 



121 



immersed weight of the tmit . ar ea of ied material is 

m (— ^ — )g. This immersed weight of sediment can be divided 

Pg-P 

into two parts m^ and where the symbol m' = m(-^ ). 

The b.ed load m^ is that part of the load which is sup- 
ported wholly by a solid-transmitted stress m^g and the 
suspended load m' is that part which is supported by a 
fluid-transmitted stress m'g. The total transport rate for 
the immersed mass of material per unit width of bed is 

H = "bs ' ^ss h ^^-^^^^ 

where 

iT-j^ = average velocity of the sand particles in the bed 
load 

and 

U = average velocity of the sand particles in the 
suspended mode of motion 

i^ = bed load immersed weight transport rate per unit 
width bed 

i = suspended load immersed weight transport rate per 
^ unit width bed 

The dynamic transport rates i^ and i^ have the dimensions 

and quantity of work rates, being the products of weight 

force per unit bed area times velocity. As these quantities 

stand, they are not in fact work rates though for the stress 

is not in the same direction as the velocity of its action. 

The dynamic transport rates become actual work rates 

when multiplied by conversion factors and A^ , each 

defined as a ratio 



122 



tractive stress needed ±q Tnainj-^-iTg transport of the load 
normal stress due to immersed weight of the load 

(4.110) 

For bed load the factor A^^ is a friction coefficient 
tan(}>, where tan(j) expresses the ratio between normal stresses 
on the bed and lateral stresses opposing motion parallel to 
the bed (which is some portion of the normal stress as in 
solid physics). The forces acting on a unit area of bed are 
as shown in Figure 4.5. 

The analogy in solid physics can be expressed with a 

solid body on a flat surface. The coefficient of solid 

friction at rest is most easily measured by resting one body 

upon the other and increasing the angle of inclination of 

the shear plane until shearing begins by gravity. Then 

since gravity force parallel to slope = mg sine}) , and 

gravity force perpendicular to slope = mg cos^ , the coeffi- 

s 

, parallel force ^ a j. j. ■ c 

cient — —f- — -r-. =. ^ = tan({) = static angle of repose 

perpendicular force s ^ ^ 

of sand. 

In solids the dynamic coefficient of friction is lower 

than the static coefficient of friction; therefore, the 

expressed angle cp would not be expected to be equivalent to 

the angle of repose except for the limiting static case 

which provides an angle of repose = 3 3° for which tan(j) = 0.63. 

s 

Bagnold (45) found that the dynamic friction coeffi- 
cient or stress ratio across shear planes of an array of 
solid grains is of the same order as the static coefficient 



123 



Lift = L 



Prog 



Weight=G, 



(G3-L5) tan<^ 



Gs-Ls =Dispersive Stress due to 
Normal Grain Momentum 



G.-U 



Friction Force Resisting Movement 
of Sand Grains 



/ 



/ 



/ 



/ 



= (Gs-L3)tan<^ CDrcg 
(for Sediment Movement) 



Figure 4.5. Relation of Normal Force to Moving Force 



124 



when the grains are closely packed and also when they are 
considerably dispersed. The actual value of tan(}) was found 
by Bagnold (4-5) to be a function of a parameter for 
dispersed solids in a fluid and is analogous to a fluid 
Reynolds number where 

^s = f 

S U ^ 

and X is a linaar . spacial. concentration defined by the ratio 

X - mean diameter d of solids 

mean free distance between solids 

The bed' load work rate then becomes 

bed load work rate = m'g U, tan<J) = i, tan4> (4.111) 

b° bs b 

since the work rate must be the resistance force (in the 

direction of the moving grains) times the velocity of the 

moving grains . 

The suspended load work rate can be inferred more 

simply. The suspended solids are falling relative to the 

fluid at their mean fall velocitv w , but the center of 

s ' 

gravity of the suspension as a whole when time averaged does 
not fall relative to the bed. Thus the fluid must be lift- 
ing the solids at the velocity w . The rate of lifting work 
done by the shear turbulence of the fluid must be 

^s 

suspended load work rate = m'gw = i, — — (4.112) 

S S D ^7 

ss 

which points out that the factor A is equivalent to 



125 



w 

A = 

= U 

ss • 

When any kind of continuing work is being steadily done 
the principle of energy conservation can be expressed in 
terms of the time rates of energy input to, and output from, 
a specified system by the equation 

rate of doing work = available power - utilized power 

or in an equivalent alternative form 

rate of doing work = available power x efficiency (4.113) 
d Work 



dt 



= P 



For bed- load transport this equation becomes 

±^ tan<|) = % ' '^■^ (U.im) 



or 



e, 

= :rf—r (4.115) 

b tantp b 

with e^^ an efficiency factor for bedload (as defined in 
Bagnold (45)) less than unity. As the bed load force used 
to determine bejilload work rate is in the direction of trans- 
port, the available power P^ for bed load transport in the 
longshore direction refers to the power expended on the 
bottom in the longshore direction. 

The power used to support the suspended load is e P 

s s 

therefore, the equation for the suspended load transport 
component becomes 



126 



w 

i — = e P ■ (4.116) 

s u s s 

ss 

or _ 

U 

ss T3 

1 = e P 

3 8 W S 

S 

The proper quantity of available power to use in this case 
P is the power available for suspension of the sand, not 

5 

the longshore power as in the bed load. More will be men- 
tioned concerning this power quantity later. 

Adding components of transport, the equation for bed 
load and suspended load transport combined becomes 

e. e U 

i = i, + i = (3^-^)P, + ( ^ ^^ )P (4.117) 
b s tanc}) b w^ s 

It will be assumed that the mean velocity of the suspended 
solids U equals the mean velocity of the fluid which for a 
steady longshore current becomes V. 

It remains to discuss the available power in terms of 
wave motion and the efficiency factors for the various 
transport modes. 

Available power for transporting sand in the longshore 
direction for bedlload is that power dissipated by the bottom 
stress in the longshore direction. In the analogous case of 
riverine sand transport the power dissipated would be equal 
to 

Available stream power = x, U 



127 



where x^^ is the bottom stress per unit area of bed and U is 
the average velocity of the stream. 

For the case of wave motion in the surf zone the power 
P, becomes 

D 

Available longshore power 

for bedload transport = t^^ • v = (x) (4.118) 

where x, = the bottom stress exerted by the combined wave 
and c-urrent action ±n the longshore direction 

V = the longshore velocity 

In the case of no lateral mixing the longshore wave induced 
stress would be from equation 4.93. 
3 S 

T - _ S 2 ^ _,>,3/2 T,v, „ sina cosa , n \ 

hy " 3x 16" P^g^-* (1-K)tan3 (4.119) 

and as shown previously in equation 4.9 6 

V = ~ J- /gD^ (1-K) tanS sina^ cosa^ ~- (4.120) 
w B 

Using equations 4.118, 4.119, and 4.12 0 and invoking the 
shallow water assumption and Snell's Law, the total long- 
shore power available for bed load transport becomes 

P^(x) = |i(i)p|- . (gD)^^^ • (1-K)^tan^g( ^ ^ )^cQsa^ cosa 

w b 

(4.121) 

For the case of suspended load sand transport the 
analogy between the available power in riverine system and 
the surf zone no: longer holds . For the riverine system the 



128 



available power for suspension of sand is the same as the 
available power for bed load motion. For the surf zone 
though, complex processes act to entrain sediment into the 
flow system such as breaking waves plunging on the beach and 
creation of turbulence by surging wave action ( similar to 
"bores" in a river). In this case the available power for 
suspension is the total power expended by the waves in the 
surf zone. .Th,us-y. at- a,, spexiific .location in the surf zone 
the available power for suspension of sand is 

P = I- (Ec cosa) (4.122) 
s dx g 

where Ec represents the energy flux per foot of wave crest 

and the cosa factor resolves the power per foot of beach 

rather than per foot of wave crest. The quantity P then 

represents the onshore gradient of the onshore energy flux. 

The total sand transport due to bed load and suspended 

load is, from equation M-.115 with U = V 

s s 

e, e V 

i = i, + i = (t^^) P, + (— ) P^ (4.123) 
I h s tan(J) b w s 

s 

where now 



^hy 


(x) 


25tt 


3 

IC 


16 


8f 

w 



V(x) 

s inct 

(gD)^'^^ (1-K)^ tan^S (— — -)'^ cosa^ cosa 

b 

(4.124) 



and 



P (x) = I- (Ec cosa) = T-|<^T ^ D^^^ (1-K) tang cosa(4.125) 

S dX g 16 



129 



Upon integration of these quantities across the surf zone 
the total transport becomes (assuming the efficiency factors 
are constant across the surf zone) 



^t) s e 

I. = / (i^ + ie) = ^ir|-r) P>,(x)dx+-^ VP^(x)dx 

X o D s tanqj o d , w o s 

/•■I o s-K sin2a, e^c, cosa, 

= (K <(1-K)tang ^( b b ^ _s_b ^).p^ (1+.126) 

w f tan*' 2 w I 

w ^ s 

where 

yH, 2 sin 2a, 

and K = constant = 1.40 
w 

which can be rewritten as 
where 

/-n o s, sin2a, e„ c, cosa, 
= (<(1-K)tane) . (_b b ^ _s_^ b) (4.127b) 

^ r tan4) • 2 w 

w ■ s 

At this point it is interesting to compare equation 

4.126 with the "modified" presently used sand transport 

relationship of Reference (25) discussed earlier, 

where 

yH, 2 sin 2a, 

■n b b 

^5, 8 ^b 2 

Equations 2.1 and 4.12 7 can be equated by the relation- 
ship 



130 



tanip 2 w 

s 



CK, — ^ tan3) 

w X 

W 



K X 



(1+.128) 



Using "rough" values of efficiency factors with e^^ = 0.13 
from an average of Bagnold's (45) range of values for bed 
load efficiency in rivers and e = 0.002 "postulated" in a 
heuristic model for suspended sediment transport by waves 
(Dean (26)), and with the following assumed values for 
typical wave and sediment parameters 



tan(|) = 


0. 


63 


= 


0. 


20 ft. /sec. (d=0 


^b = 


2 


feet 


^b ~- 


8° 




f - 

w 


0. 


02 


tan3 = 


0 . 


01 


a value of 


^Ib 


can be found 


^Ib" 


0. 


50 


As noted 


in earlier, the 


sand transport 


computations is 



'ib ~ ^'^ although many inves- 
tigators (111) believe that the present value used is too 
high and should be on the order of 0.M--0.6 based on 

reanalysis of past data. 

It therefore appears that the presently postulated sand 
transport formula appears to provide a reasonable approxi- 
mation for the prediction of sand transport on beaches. 
This will be checked out further later. 



131 



4. '1. 3 Sand Transport Distribution Across Surf Zone 

Now, using the previously developed (4-6) distribution 
of longshore current velocity as given in equations 4.100 
and 4.10 8, and the presently developed sand transport 
relationship given in equations 4.12 3, 4.124, and 4.12 5, the 
distribution of sand transport in the lateral direction 
(across the surf zone) can be calculated. 

. .Again the equation for' sand transport can be written 

i . i + i = ^ . (T V) + !il 8(EPg cosa) 
X, b s tan<p hy w^ 3x 

where from equation 4.9 0 ~-^y t)^ written 

T,„ = f , (J /pT /X) V (4.129) 

hy w 2 TT ° D 

where X is — , a dimensionless distance from the shoreline 
^b 

and 

°b ^b 
^b Tl-K) tang ^ <(1-K) tanB (4.130) 

This equation can be written in terms of the dimensionless 
variable X as 

, _ ^b ^ ^^w'^b „l/2„2 

P — ^ ' 



I tan^ ^ 2¥ 

+ — (y|- c, D, (1-K) tanBcosa) X^^'^V (4.131) 

w 15 b b 

s 

Now the above mentioned longshore current model (which 

incorporates lateral mixing) as discussed in equations 4.100 

and 4.108 can be used in the above expression. It should be 



132 



noted that outside the surf zone there can still be bottom 
friction in the case of lateral shear mixing, as the bottom 
friction balances the gradient in lateral shear force be- 
tween adjacent water columns (since the gradient of the 
"radiation" stress is zero). Hence, bed load can exist 
outside the surf zone. As the available power to suspend 
the bottom sediment is considered to be due to the wave 
breaking process though, there Is no suspended sediment 
transport outside the breaking zone. 

Using equations 4.98, 4.12 3, 4.124, and 4.12 5, the 
equation for sand transport as a function of the distance 
across the surf zone can be written as (after considerable 
manipulation and use of the shallow water assumption, along 

V 

with a dimensionless longshore velocity defined as V = tj— ) 

B 

e, sin2a, i m 
^% Han(i)-2 

e^c^cosa -^3/2^ ^25tt^ ^ K:(l-K)tanP ^ ^ (4.132) 

w 16 f x, 

s w b 

which can be written in dimensionless form as 

777TT£K7tanF7 ^16 ' ^ tan(j)-2 

f ^ 
w 

e c, cosa, a. ^ / o 

^ V X^'^^} (4.133) 

w 

s 

The distribution of sand transport across the surf zone 
can now be found using assumed values of the efficiency 



133 



factors and other appropriate wave and sediment parameters. 
As an example the following values are considered: 
e, = 0.13 

D 

tancj) = 0.63 



W : 
S 


■ 0. 


20 ft. /sec. 


e : 
s 


: 0. 


002 




: 2 


ft. 


'- 


: 8° 






: 0. 


2 



A dimensionless plot of the sediment transport lateral 
distribution for the above assumed values is shown in Figure 
1+.6. As referred to previously, there is no allowance for 
suspended sediment transport outside the surf zone in the 
present model since the energy dissipation (available power 
for sand suspension) is zero outside the region of breaking. 
This leads to discontinuity in the sand transport distribu- 
tion at the surf zone. The question as to whether such a 
distribution occurs in reality cannot be answered as no data 
existjwith which to compare such distributions although at 
least one researcher, Ingle (75), found that little sand 
transport occurred outside the surf zone during flourescent 
sand tracing studies. Other researchers (73, 7 4) make 
reference to the absence of sand transport seaward of the 
wave breaking area based on laboratory results although no 
quantitative data aire given . From color aerial photography 



134 



of the surf zone it also appears that the breaking wave zone 
provides a clear demarcation between the "colored" water 
suspended load area inside the wave breaking zone and the 
"clear" water area outside the wave breaking zone which is 
at least qualitatively in agreement with the present model 
(i.e. lateral mixing of sediment seaward of surf zone is 
negligable). 

Using the postulated sand transport lateral distribu- 
tion, the sand transport can be integrated across the surf 
zone (assuming a mixing coefficient value) and a "refined" 
value of coefficient can be found. Again using equation 
4.131 and integrating across the surf zone a dimensionless 
equation is found; 



, 25tt , 

K(l-K)tan6^ 16 ^ 



•) 



e, sin2a, , 
tancp • 2 



w 



(4.134) 



e c,cosa, , 



w 



which can be rewritten as 



I 

^ ic(l-K)tang ^ 

f ^ 
w 



(^) 
^16 ^ 



e, sin2a, .,. ^ e c, cosa, j' 

(-h ^)(i" + t") + ( s ^ ^ ) i" 

^tant})-2 ^^^i^^2^ ^ w ^ ^3 



(4.135) 



where 



135 




136 



I* = /^V^X^'^^dX (4.136) 
o 

I* = TV^X^/^^X . (4.137) 

^ 1 

I* = /^VX^'^^dX (4.138) 
o 

using the solutions for V presented in equations 4.100, 
these integrals can be integrated to provide the following 
values 

for "p" arbitrary (but i 2/5) 

* B^^ 2AB^ 2A^ 

^1 = (ip^+3/2) ^ (p^+5/2) ^ T" (4.139) 

ft -"^2 

^2 ^ 2p2+2 (4.140) 

for "p" = 2/5 
ij = 0.0594 

I2 = 0.0139 (4.142) 
I* = 0.1166 

The values of these integrals have been tabulated and 
are presented in Table 4.2. 



137 



Table 4-. 2. Integrals for Sediment Transport. 







A 




A 


p 






V^2 


T 

^3 


• UU± 


OC A A 


A A O C 


A C A C 

. 2635 


A T A C 

. 2706 


• Quo 


A 1 1 A a 

. 24-28 


A A C O 

.0058 


A P 1 O 

. 2486 


. 2603 




A 1 O T 

. 2137 


A A A A 

. 0092 


A O O A 

. 2229 


A 1 1 T A 

. 2419 




.1543 


AT 1 ) 1 1 

. 014-4- 


. 1587 


A A A C 

. 2005 


-10 


T A A O 

. 1228 


. 0158 


n o A c 

. 1386 


.1758 


. 15 


. 1037 


.0160 


.1197 


. 1596 


. 20 


A A A A 

. 0902 


A T C A 

. 0158 


1 A A A 

. 1050 


.1474 


. 25 


A T n n 

. 0799 


. 0154- 


. 0953 


.1377 


. 30 


. 0718 


. 014-9 


A A 

. 0867 


.1296 


.35 


.0651 


. 014-4- 


. 0795 


.1225 


. 40 


.0596 


. 0139 


. 0735 


.1167 


. 45 


.054-7 


. 0134- 


. 0681 


.1113 


. 50 


.0505 


. 0129 


. 0635 


.1055 


. 55 


A 1 1 T A 

. 04-70 


. 0124- 


A C A 1 1 

, 0594 


T AAA 

. 1022 


. 50 


. 04-09 


. 0115 


. 0524 


. 0948 


. 70 


. 0384- 


. 0110 


. 0494 


A A T C 

. 0915 


. 75 


. 0361 


AT A 

. 0105 


A 1 r T 

. 0457 


A A O C 

. 0885 


. oO 


A A 1 1 A 

. 0340 


A T A A 

. 0103 


. 0443 


A O C "7 

. 085 / 


. 85 


. 0322 


. 0099 


. 0421 


. 0831 


.90 


.0304- 


.0095 


.0399 


.0806 


.95 


.0289 


.0092 


.0381 


.0783 


1.00 


.0274 


.0089 


.0363 


.0752 


1.20 


.0227 


.0078 


.0305 


.0687 


1.50 


.0175 


.0064 


.0240 


.0601 


3.00 


.0071 


.0031 


.102 


.0373 


6.00 


.0024 


.0012 


.0035 


.0214 


10.00 


.0010 


.0005 


.0015 


.0137 



138 



Estimation of .£f f i ciBiicy Factors and Dynamic Friction 
Angle 

It remains to discuss appropriate values of the effi- 
ciency factors e^ and e^ for the case of sand transport and 
the dynamic coefficient of friction for bed load movement. 

The suspended sediment efficiency factor e^ has been 

defined previously as 

work rate necessary to maintain a suspended load 
s available wave power per foot of beach 

(14.143) 

S 



3x —5- • c • cosa) 
0 g 

= Ss-1 

where m' of equation 4.109 = C m^ (— f — ) 
s ^ f Sg 

and C = time and spacially averaged (over depth) dry weight 
concentration of sediment 

mjT = mass of fluid in a unit area of surf zone at a given 
depth = pD 

Thus e^ can be reduced to the following equation 

e = ^ s (4.iLm) 

S o 2 

c cosa(l-K)tanS 
16 g 

To determine a value of the factor e^ the problem becomes 
one of postulating a method for determining the time and 
spacially averaged concentration over depth using wave and 
sediment parameters. Heuristic models will first be pre- 
sented to define the important parameters and then a 



139 



semi-empirical approach based on existing observations of 
suspended load in waves will be used to formulate a model 
for finding the concentration. 

Assume first that a breaking wave crest suspends mater- 
ial to a maximum height in the water column which will be 
considered proportional to the breaking wave height = 6H as 
shown in Figure Now, if the time required for the 

grain to fall back to the- boirtroTii -t. = — is less than the 

b w 

s 

wave period T, then the depth averaged concentration as a 

function of time, as well as the depth and time averaged 

concentration will be shown by Figure 4. 7b. If the time 

required for the grain to fall back to the bottom t^ is 

b 

greater than the wave period (where the dominant crest force 
is responsible for suspending the sediment) the concentra- 
tion will be shown by Figure ^-.Ic, with higher depth-time 
averaged concentrations than the previous case. It is 
therefore apparent that the depth time averaged concentra- 
tion should be a function of the parameter This 

^s 

heuristic argument should be kept in mind as a more for- 
malized approach is now used in an attempt to arrive at an 
important parameter for use in prediction of suspended sand 
concentration in the surf zone. 

It is important at this point to look at the findings 
of some investigations of suspended sediment in the surf 



140 





Figure 4.7. Sediments Suspension Due to (b) Long Periods, High Fall 
Velocities, (c) Short Periods, Low Fall Velocities 



141 



zone. Watts (112) conducted field studies using an elabo- 
rate continuous suspended sediment sampler from a pier. His 
results showed that the amount of sand in suspension was 
related to the wave height, or energy, of the waves for a 
particular test. In these experiments, and some by Fukushima 
and Mizoguchi (113) using suspended samplers made of bamboo 
poles, the vertical distribution of suspended sediments was 
also measured. These data showed that the amount of sus- 
pended sediment can be fairly well described over the ver- 
tical by plotting concentration versus elevation on a semi- 
logrithmic plot for the case of spilling breakers. Fair- 
child (IIM-) using field equipment similar to that of Watts 
and measuring suspended sediment from piers found results 
similar to Watts. In most all cases where plunging breakers 
were not encountered, the concentration of suspended sedi- 
ment was found to be exponentially distributed over the 
vertical water column. Fairchild (114) also found this 
exponential type distribution in similar studies conducted 
in a wave tank. Hom-ma and Horikawa (115), Shinohara et al . 
(116), Hattori (117), and Kennedy and Locher (118) have all 
found this exponential type distribution under shoaling 
waves which is suggestive of a constant "sediment exchange 
coefficient" in the vertical. This appears to be particu- 
larly the case for spilling breaking waves (116). Reviewing 
the equilibrium exchange equation for sediment in the 



142 



vertical water., column- as given Ro.ose (64) where (using 
time averaged values as denoted by a single bar over the 
quantity, and primes as fluctuating quantities) 

w C(z) = -£ = V C (4.145) 

s s dz p 

where £ = vertical sediment exchange coefficient analgous 
s 

to the momentum exchange coefficient or eddy viscosity, and 

is the sediment particle random velocity. If the exchange 
coefficient £^ is independent of the z direction, then upon 
integration an equation can be found of the form 



C(z) J ^s^^-^a^ 

exp < - 

Ciz) 

a. 



£ 

S 



(4.146) 



where C (z ) is a concentration at the level z above 
a a 

the bottom. Since z is an arbitrary distance it. can be 

a 

assumed that z^ corresponds to the top of the bed load zone 
(Z = 0) and the concentration C (z) can be depth integrated 
from z = 0 to z = D giving 

_ -w D £ 

C = C (0) (1 - e — ~) —fr (4.147) 

£ w D 

s s 

A method for attempting to rationalize out a parameter 
of importance in suspending sediment ( "entrainment" ) to a 
level near the bed (in order to empirically obtain a value 
of ^(0) and hence C) is to consider the forces on a grain of 
sand in the near vicinity of the bed about to be lifted out 
into the flow (see Figure 4.8). 



143 



The lift force on a given grain "L" can be expressed as 

S L Z 4 

and the gravity force weight G action on the particle is 

s 

expressed as 

3 3 
= (p^ -p) g = pCSg - 1) g (4.149) 

Now the ratio of lift to weight is 

2 



L 3C, u, 

s L hw 



G_ " 4CSg - Dgd 



(4.150) 



As u^^ is oscillatory, the time average value of the lift to 
gravity is 

L 3C, u, ^ 

G- - 8(Sg - Dgd C4.151) 
s 

In linear wave theory the maximum velocity (just outside the 
wave boundary layer) is 

u, (-D) = I ( ' I ,r. ^ (4.152) 
hwm T smh kD 

and the lift to gravity ratio can then be expressed as 



r 3c, 

s _ L 



H IT 

T sinh kD 



^ =^ (4.153) 

G 8KSg - 1) gd 

o 

As grain size pertains to spherical particles in the 
above formulation a more general formulation applying to non- 
spherical particles can be made by expressing the grain size 



144 




Figure 4.8. Vertical Forces on a Grain Particle 
About to be Lifted from the Bed 



145 



in terms of its fall velocity w by a balance of drag force 
to submerged weight of the grain. This balance gives the 
following equation for a spherical particle falling at its 
terminal velocity 



w = (i - 1^ gd)l/2 

"s ^3 ^ 



(4.154) 



where is a drag coefficient. Now equation 4.15 3 can be 
rewritten as 



2C 



D 



H TT 

T sinh kD 



(4.155 ) 



w 



and thus the parameter of importance in predicting a concen- 
tration C(z ) in the vicinity of the bottom is the ratio of 
a ^ 

bottom horizontal water particle velocity to sediment par- 
ticle fall velocity 



IT 



C(z ) = function ■ 
a 



H 

T sinh kD 



w 



(4.156) 



In shallow water, sinh kD Z kD and the above expression can 
be reduced to 



C(z ) = function 

3. 



w ^D 
s 



Note that in equation 4.15 6 the parameter 



H 



(4.157) 



heuristically 



arrived at previously, appears. 



146 



To obtain the relationship of e to wave parameters a 

s _^ 

look at the basic physical mixing phenomena is again neces- 



sary. The mean product V' C can be correlated and the 

ST 

degree of correlation expressed by a correlation coefficient 
3^ which can be defined as 



3 = 2 (4.158) 

/ ,/ 

9 7 
C ' V 
P 

Then analogous to turbulent flow theory (54) the flue- ' 

tuatmg concentration y C ' can be related to the mean concen- 
tration gradient as 



C'^ = (4.159) 

where 5-^ = a mixing length analogous to the mixing length of 
Prandtl (64) for turbulent flow. Using the above two equa- 
tions a new equation for the turbulent fluctuation can be 
found as 

= - ISi! V'y^ • h§ (4.160) 

where the minus sign expresses the fact that the transport is 
in the direction of decreasing concentration. 



The product ^^\J ^ ^'^ is known as diffusion coefficient 
for sediment and is given the symbol e^. 



147 



where l-^ is in length units, is dimensionless , and ]J V^^ is 
the root mean square fluctuating velocity component. 

It will now be assumed that in spilling breakers the 
turbulence level is constant over depth and that the mixing 
velocity scale is proportional to the turbulence velocity 
which, in turn, is proportional to the average energy dissi- 
pation in the water column averaged over depth. 



I— (Ec cosa) 



- ^ ^ (4.162) 

Using shallow water assumptions, linear wave theory, and 
assuming as before that wave angles are small the following 
equation can be found 



Now, the mixing length l-^ will be assumed proportional 
to H which is the vertical distance a water particle on the 
surface travels through during the passage of a wave. 

H-^ " E (4.164) 

The sediment mixing parameter can then be found from 
equations 4.151, 4.16 3 and 4.164 to be (with = constant) 

£ = constant ^ ;g '^^^^ (4.165) 



148 



The data to be used in the assessment of the parameters 
C(0) and e come from field studies done by Fairchild (119) 
and Kana (120). In the study of Fairchild, suspended sedi- 
ment concentrations were taken throughout the surf zone from 
a pier at Nag's Head, North Carolina. The concentrations 
were measured using a tractor mounted pumping device attached 
to a pipe manifold extended over the pier into the surf 
ione. Time average concentratioixs were taken and the concen- 
tration was measured as the dry weight of suspended sediment 
filtered out per weight of fluid pumped. 

In the measurements of Kana, the concentrations were 
obtained at a location on the beach near Price Inlet, South 
Carolina by individuals stationed throughout the surf zone 
using water samplers activated by the individuals. The 
devices are described in (120). The concentration is then 
the weight of suspended sediment per weight of water sample. 

Data used consisted of depth-concentration profiles for 
spilling breaking waves in the surf zone. Wave parameters 
were measured by a wave gage at the end of the pier in the 
case of the Fairchild data, and by visual wave estimates 
(using wave poles and stopwatches) in the case of the Kana 
data. Unfortunately beach slope at the data collection 
sites was not taken. In most all cases the data gave reason- 
able results with the assumption that the mixing parameter 
is constant throughout depth, that is, the data plotted up 



149 



as straight lines on a semi-logrithmic plot of concentration 
versus elevation above the bottom. Data sources and data are 
given in Appendix B. The data used were for spilling break- 
ers although in a few cases it appeared that the concentra- 
tions were scattered with depth (not straight line) in some 
of the questionable type breaking wave data. In these cases 
it was assumed that the data scatter was due to plunging 
type breakers and the data were discarded . 

From the data plots and the sediment property analysis, 
values of were calculated and values of C(0) obtained 
from plots. These values were then plotted as functions of 
the relationships provided by equations 4.157 and 4.165. 

The results of the analysis are shown in Figures 4.9 
and 4.10. Figure 4.9 shows that in fact the sediment mixing 
coefficient is well represented by a linear trend with the 

parameter - — g — which is a measure of the turbulent 
energy dissipated throughout the water column modified by a 
mixing length Cin this case wave height). Figure 4.10 
provides a functional relationship between C(0) and a par- 
ticle lift to weight ratio parameter. At lower values of 
the lift to weight ratio parameter the concentration C(0) 
drops off sharply while at higher values there appears to be 
an approach to a linear trend. More data would be needed to 
confirm this trend outside the limits of the existing data. 

The bed load efficiency factor is found from argu- 
ments postulated by Bagnold (45) concerning a movable 



150 




151 



+ 




S a. Q 

i£2 X $ 



o 



c 




o 




•r- 




+-> 




fO 




S- 


s- 


-(-> 


0) 




+J 


OJ 


O) 


u 


5= 


c 


(13 


o 


S- 


o 


ns 




a. 


-u 






o 


OJ 


■r— 




-P 


•r- 




-o 


q: 






00 












<u 




0) 






+j 




OJ 


o 


CQ 




Q-+-> 


•r— 


<+- 


J= 






_i 






O T3 


•r— 




+J 




n3 

















^ o 
ro ^ 

•1— 

Li- 
eu 



152 



boundary. Bagnold (US) pos-tiilates a continuous moving grain 

carpet relative to a stationary bed. Assuming a velocity of 

the moving carpet = the shear on the carpet (assuming 

2 

fully turbulent flow) is t ~ (V-V^) . The transporting work 

2 

done is then xV " V CV-V ) . This function has a maximum 

c c c 

value when = ^ hence the maximum transport efficiency is 

TV , 
c _ 1_ 

®c " tV 3" 

Bagnold then postulates a similar slip of the actual 
grains within the moving grain carpet relative to the carpet 

which introduces an efficiency factor e = jj— such that 

g 

e 

e, = e • e = Again the work rate on the moving grains 

b c g 3 ^ 

n ' 

has a value V, (V -V, ) where n' varies between 1 and 2 
b c b 

according to the local grain Reynolds number (V^-Vj^)d/v, 

being 1 in the Stokes law region and 2 for large grains . 

As before the work rate has a maximum when V, /V = 

b c 

l/(n'+l). The exponent n' for a given grain size d and slip 
velocity of grains V^-V^ can be obtained from the slope of 
the experimental log curve of the sphere drag coefficient 
versus the local grain Reynold's number. Values of e and 
e^ can then be found corresponding to a mean flow velocity V 
for a given grain size. As per Bagnold (45), the values of 
e^ and e^ are given in Figure M- . 11 over an extended range of 
values of V. 



153 



The dynamic Bedlload friction factor tan^) has been 
mentioned previously to be dependent on solid/fluid param- 
eter analogous to a fluids Reynolds number. Bagnold (121) 
found that tan<i> was dependent solely on a parameter 

d /v^ 

~ where as before, X = ratio of the mean diameter 

^ ^ ^ X 

of the solids to the mean free distance between the solids 

a-nd p - d-en-slty of -the solids. 

The parameter R^ is thus analogous to a "shear" Reynolds 

number of the form 

^ _ d /t/p _ du^ 

shear ~ y / p " v 

In experiments using grains of equal density to the 

fluid being sheared, Bagnold (121) measured normal and 

tangential stresses due to the grain collisions in the 

annular space between two concentric cylinders as a function 

of the parameter R . The results of this series of tests 

s 

are given in a plot by Bagnold (121, 4-5) and reproduced here 
as Figure 4.12. The value assumed for X was equal to 14 
which was the limiting concentration at which a sheared 
array of solid grains was found to cease to behave as a 
Newtonian fluid and to begin to behave as a rheological 
"paste" (121). 



154 




o d o d 



155 



CVJ 
T3 



CM 



o 



8 



<M 



O 
O 



g 



o 



CO 

O 



d 



d 



a 



o 



UDj 



rO 

d 



156 



4.4.5. Model Comparison with Laboratory and Field Results 

In a comparison of the model with existing sets of data 
for sand transport, four data sets were chosen, two sets of 
laboratory data and two sets of field data. Although there 
are many existing data sets as discussed in Chapter II, 
other data sets failed to include vital wave parameters 
necessary to the model comparison; most typically, breaking 
wave, .ajigla was^nat given -but ojily the offshore wave angle 
measured at the wave generator (in the case of laboratory 
data) or in deep water (in the case of field data). It was 
felt that transforming wave parameters to breaking condi- 
tions involved inclusion of an additional uncertainty factor 
in the data and was therefore undesirable from the stand- 
point of assessing model validity. 

The four data sets chosen for model assessment include: 
the Shay and Johnson tests (laboratory); the Fairchild tests 
(laboratory); the Watts tests (field); and the Komar tests 
(field). Parameters were calculated in accordance with 
equations, graphs, etc., of this chapter and the calculated 
model parameters are listed in Appendix B. Wave heights are 
assumed as root mean square wave heights in the case of all 
data. In the Watts data, model parameter values were 
averaged on the basis of 0.5 to M- days of wave records, the 
time interval in which the sand transport values were given. 

As was shown earlier, the fact that the longshore 
current is not linear across the surf zone causes the value 



157 



of in equation 4.126 to be different than the postulated 
value. 

Also, due to the randomness of waves in nature, it 
might be expected that a considerable portion of the break- 
ing waves might be of a plunging type though the average 
wave conditions are of a spilling breaker type. Kana (12 0) 
has noted considerably higher concentrations of sediment in 
the plunging typ^ breaJcers which would predict much higher 
suspended load and, in turn, higher total sand transport. 
For these reasons, the sand transport model correlation was 
made plotting the immersed weight sand transport I,^ with the 
factor XP^- 

The plot of all four sets of data which cover-- four 
orders of magnitude of the 1,^^ range is given in Figure 4.13 
by the best fit equation: 

= 2.82xPj^ . (4.166) 

A more detailed plot of the Fairchild data alone is shown in 
Figure 4.14 where the best fit equation to Fairchild data is 
given as : 

= S.OlxP^^ 

The best fit equation to the field data of Komar and Watts 
is given by the equation: 
I, = 4.24XP, 

and is shown in Figure 4.15. 



158 




Figure 4.13. Sand Transport Model Relationship 



159 



I^(#/sec) 




Figure 4.14. Best Fit Sand Transport Model for Fairchild Data 



160 




Figure 4.15. Best Fit Sand Transport Model for Field Data 



IGl 



As assumptions .liav.e ieen incorporated into the model 
which cannot be checked out by the present data (i.e., 
roughness, spilling breaker assumptions, monochromatic wave 
assumption) the model appears to provide a reasonable fit. 
Additional supporting evidence for the bed load component of 
the sand transport model is given in the work of Barcelo ' 
(82) who ran tests as noted in Chapter II in which he noted 
that no component o-f -suspended load was present below a 
critical wave energy level. For wave energy levels below 
that critical value (bed load), the sand transport was found 
to be proportional to the square of the sine of the breaking 
wave angle x 2 , that is : 

Q. ~ sin^(2a- ) 

Conclusive proof of the postulated model must await detailed 
laboratory testings in large wave basins with measurements 
made of all necessary wave, beach, and sediment parameters. 



CHAPTER V 

APPLICATION OF SAND TRANSPORT EQUATION 
TO CALCULATION OF LITTORAL DRIFT USING SHIP WAVE DATA 

5 . 1 Application of Model 
In this chapter the equation derived for sand transport 
will be applied to compute littoral drift using a source of 
visual wave observations made from ships. 

Prior to application of the sand transport equation it 
is worthwhile to review some of the important assumptions 
inherent in the method to be used. 

CD Linear theory is valid for the wave transformation 
process and the wave energy present in the wave 
system; 

(2) Assumption in calculation of a "friction" 

coefficient are not violated Csee Reference 59); 

C3) Bottom topography is composed of straight and 
parallel bottom contours; 

CU) No drastic changes in the bottom profile are 

encountered in the shallow areas seaward of the 
breaker line up to the beach; 

(5) Adequate sources of sand are available for trans- 
port; 

Item (1) refers to the mathematical formulation of the 
problem and its relation to physical reality. As has been 



162 



163 



previously noted, this assumption is reasonably good up to 
the region of breaking waves where it departs from the 
actual situation. Item (2) assumptions will be discussed 
later. Assumption (3) is necessary for the simple applica- 
tion of Snell's Law of Refraction used in this report and 
does not require a monotonic decrease in depth toward shore, 
but only the aforementioned relationship between bottom 
contours. Assumption (4) is necessary due to the use of 
offshore wave conditions for the computation of longshore 
energy rather than nearshore conditions. Thus, rock or 
coral reef might cause a large dissipation or reflection of 
energy before the wave reaches the computed breaker zone, 
which would be incorrectly included in the estimation of 
expended wave energy in the surf zone. An additional assump- 
tion inherent in the same transport model is Item (5), the 
availability of sand to be moved- This is dependent on the 
geologic processes acting in the area, and the natural or 
man-made conditions present. In some shoreline areas of 
Florida there is a lack of sand, predominantly in areas 
having extremely low wave energy. Rivers, inlets, jetties, 
groins, seawalls, prominent headlands, and submarine ridges 
and valleys can also cause a lack of sand in an area downdrift 
of an obstacle. A lack of sand supply causes erosion and in 
turn a depleted sand reservoir, with less sand available for 
the transport downdrift of a barrier. 



1614 



The littoral drift , can -be e:xpressed in terms of a 
volume transport rate rather than immersed weight transport 
as noted in Chapter II by the following conversion 

I, = (p^-p) g(l-p^) Q, (5.1) 

Due to the present more popular method of expressing the 
transport rate as a volumetric rate, the results of this 
chapter will provide values of (volumetric transport 
rate) in cubic yards per day. 

5^ 1:1 r Pata "Source 

The wave data used in the computation of longshore 
energy flux and consequent littoral drift in this report can 
be found in the U.S. Naval Weather Service Command — Suimnary 
of Synoptic Meterological Observations , Volumes 4 and 5 
(122), hereafter referred to as SSMO. These volumes are a 
compilation of meterological and sea state observations 
taken from ships travelling through "Data Squares" defined 
by their latitude and longitude boundaries. The percent 
frequency of wind direction versus sea heights can be found 
in SSMO Table 18 for different data squares on a monthly and 
annual basis. The percent frequency of wave height versus 
wave period for both sea and swell observations can be found 
in SSMO Table 19 for different squares on a monthly and 
annual basis. Computations of sand transport will use the 
data from both of these tables. Necessary assumptions made 
in the use of SSMO data are presented and discussed below. 



165 



In the use of Table. 18 the., assumptions have been made 
that (1) swell waves are in the same direction as the sea 
waves, which in turn correspond to the wind direction; and 
(2) waves are propagating in one direction only, the ob- 
served direction, in any specific time interval. In apply- 
int Table 19, the assumptions are made that (1) sea and 
swell waves of the same period and height can be treated 
alike, and will not lose- energy to the atmosphere between 
the point of observation and the portion of coastline con- 
sidered; (2) no other wave heights or periods af^e present 
during the observation of recorded wave with a given height 
and period; and (3) all observations were made in" "deep 
water" (h>_2.56T^ in ft.) for the wave periods recorded. 

Correlation between the ranges of wave heights, peri- 
ods, and directions given in the SSMO data volumes and the 
corresponding values used in the calculations of drift can 
be found in Appendix II. Due to the nature of human obser- 
vation of waves, the heights and periods found in the data 
tables should be considered as-significant heights and 
periods . 

5.1.2 Analysis of Wave Data to Compute Sand Transport 

The immersed weight sand transport rate as given by 
Equation M-.12 7 can be modified to provide the volumetric 
sand transport by the equation 5.1 which can then be ex- 
pressed as 



166 



YCS -1)(1-p ) 



(5.2) 



where the terms x and depend on wave climate parameters 
H^, T, as well as sediment fall velocity. The above 
equation represents sand transport in terms of one wave 
height, period, and direction, in a deterministic sense. 

Considering a continuously changing state of offshore 
wave .conditions-, heights-, .periods-, ..and directions , the total 
littoral sand transport would consist of average sand trans- 
port weighted in accordance with values of for represen- 
tative wave heights, periods and directions. Thus for 
continuously changing wave conditions , the total longshore 
sand transport as averaged over a time interval t* would be 



^ ^ Jt=o Jt=o ^ 



The value ^ can be thought of as the fraction of time over 
which a specific wave having a certain height, period, and 
direction is being generated during the period t". Express- 
ing these results in finite intervals : 

f(H^,T,9) = frequency = ^ (5.4) 

and 

t=t* 

Q = I Q/t).f(H^,T,e) (5.5) 
t=0 



1B7 



where 

F =00 

O T = oo e = 2TT ^ 

I I I f(H^,T,0) = 1.00 C5.6) 

H =0 T=0 9=0 
o 

with 6 equal to the azimuth of the direction from which the 

wave is propagating. It is related to by the equation: 

a = 0 - 9 where 9 is the azimuth of the perpendicular to 
on n 

the shoreline (see Figure 5.1). 

For waves reaching the coast, the summation would be as 
follows with 9 = 9^ - ct^ and ranging from -90° to +90°; 

H „ 9=9 +5 

0 T=<=° n 2 ^ 

1 I I ^ f(H T,9)<1.00 (5.7) 

H=0 T=0 9=9-4 
o n 2 

Note that in the above summation, when waves are being propa- 
gated away from the coast, that no wave energy will be avail- 
able for sand transport. Therefore the total sand transport 
becomes 

Qo = I I l"" ^ Q (H ,T,9) . f(H T,9) (5.8) 

^ H =0 T=0 9=9 -1 ^ ° ° 
o n 2 

The value of f(H^,T,9) can be computed by means of SSMO 
Tables 18 and 19. From Table 19 a value of f-^g(H^,T) is ob- 
tained such that 

H =" 

^ig^H^'T) = 1.00 (5.9) 

H =0 T=0 
o 

From Table 18 a value f-,o(H ,9) can be obtained corresponding 

X o o 

to a wave height range in Table 19 such that 



158 




Figure 5.1. Definition of Azimuth Angle Monnal to Shore e^, 

and Azimuth Angle of Wave Propagation e 



169 



f,o(H*,e) = 1.00 (5.10) 
6=0 1^ ° 

where the * represents the correspondence of in Table 18 
to the same range in Table 19 . Multiplying these two fac- 
tors together gives the desired frequency as a function of 
wave height, period, and direction. 

?(H^,T,e) - ?^g.?^g (5.11) 

By the use of Equation 5.8, the longshore sand transport 
can be obtained in cubic yards per day, as averaged over any 
given period of wave observations. As mentioned previously, 
the representative values of H^jTjB for the ranges given in 
SSMO are discussed in Appendix C. 

The procedure for the calculation of longshore sand trans- 
port Q,(H ,T,a )«f(H ,T,9) is as follows: 
X. o o o 

(1) Compute the onshore directed component of energy 
flux 



' o 

c cos a. 



from deep water conditions, that 



8 go o 
is, the representative conditions for given wave 
height, period, and direction ranges (where 

E = — ^) 
^o 8 ^• 

2 

(2) Compute the quantity , a bottom frxction energy 
dissipation coefficient discussed in Chapter II, to 
a shallow water depth, h , outside the zone of 



170 



breaking waves by numerical integration procedure 

of reference (59) (along the coast of Florida this 

depth was normally taken as 10 feet). 

(3) Calculate the breaking wave angle by an equation 

based on Snell's Law and linear wave theory as 

postulated by LeMehaute and Koh (12 3) 

a, = a • (0.25 + 5.5 E/h) (5.12) 
bo o o 

- my Calculate- the -co efficxHut "x and the longshore 

energy flux P^^ (as modified by friction dissipation 

(59) where 

2 2 
P = Kj:,(E, c ,cosa, )sina, = K^(E_c _cosa^)sina, 
Jifbgb b b fogo o b 

(5.13) 

(5) Find f^g and f^g values in SSMO Tables 18 and 19 
as mentioned previously, and calculate f=f^g'f^g. 
Calculation of is then a simple summation process 
in which the data must be put through a "filter" to elimi- 
nate all differential bits of sand transport with azimuth 
directions 0 that are less than -9 0° or greater than +9 0° to 
the coastline azimuth 6^. When looking offshore a positive 
values of = Qi^^^ is recorded for waves propagating from 
the left side and causing longshore sand transport to the 
right; and likewise, a negative value of Q^^ = is re- 

corded for waves propagating from the right and causing 
longshore sand transport to the left (see Figure 5.2). 
Summing the positive, negative, and total values of long- 
shore sand transport gives the quantities Qjj^+j ^Jl-' "^ilnet * 



171 




Figure 5.2. Relationship Between Direction of Wave Propagation 

and Direction of Longshore Sand Transport 



172 



Additional assumptions used in the preceding method of 
calculation which were not previously discussed are: 

(1) There is minor loss of energy through bottom fric- 
tion between h^, the shallow water depth at which 

is calculated and the breaking depth. 

(2) Computation of K^, , and involve linear 
theory and inherent assumptions. 

(3) i_s ca.lculated -usi-Qg a bottom profile perpendicu- 
lar to the stretch of shoreline considered rather 
than the actual profile over which the waves 
travel. Inherent in this procedure is an addi- 
tional assumption that the wave climate used 
occurs at a point offshore perpendicular to the 
portion of coastline considered. 

(M-) Data weighting of wave heights, periods, and 

directions is accomplished linearly using the 

centroid of the SSMO data square as the source of 

the offshore wave data (see Figure 5.3 for location 

of SSMO data squares). 

(5) Assumed friction coefficient f = 0.02. 

w 

5 . 2 Results of Sand Transport Computations 
Littoral drift "roses" with annually averaged values of 
littoral drift in cubic yards per day have been computed 
using the SSMO annual data summary tables along sections of 
Florida's sandy shores. These are presented in Figures CI 



173 




174 



through C26. Because of the large number of these figures 
they are located with the Appendices Section of this thesis. 
A littoral drift rose diagram for each section of coast 
considered gives the annually averaged total positive and 
negative littoral drift. Positive values of littoral drift 
refer to drift moving toward the right when looking offshore, 
and conversely, negative values of drift are quantities of 
drift moving to the. J. eft a^, noted previously. On the East 
Coast of Florida a positive value of drift would thus repre- 
sent Southward drift, while on the Gulf Coast, the reverse 
would be true; that is, a negative value of drift would repre- 
sent Southward drift on the Gulf Coast. The net drift values 
represent the difference between the Southward and Northward 
total values of drift with the direction of the drift indi- 
cated by its sign as described above. Although the littoral 
drift has been computed for coastline orientations ranging 
over 360° of the compass, in actuality, the coastline orien- 
tations range at most over 18 0° for any given section and 
have been presented showing the maximum practical range plus 
or minus 20° for local anomalies. 

As mentioned previously, these values of littoral drift 
are for stretches of coast exposed to the ocean wave climate 
as represented by SSMO data. They are not valid for bays, 
lagoons, or estuaries, where the shoreline is not exposed to 
a wave climate represented by the SSMO data. Also, they are 
not valid where local anomalies exist in offshore bathymetry. 



175 



5.2.1 Use of a Littoral Drift Rose 

Use of a littoral drift rose is as follows: 

(1) Determine the orientation of coastline at 
which a drift is desired. 

(2) Using the azimuth of the seaward directed normal 
to the coastline at the location, find the values 
of total positive and total negative littoral 

, ..drift assoaia.ted with thi.^ azimuth angle on the 
proper drift rose corresponding to the desired 
location . 

(3) Find the value of net drift as the difference 
between the positive and negative drift values. 
If the net drift value is positive, the net drift 
will be to the right when looking offshore; if 
negative, the net drift will be to the left. 

To demonstrate the method, values of net drift at Ponte 
Vedra Beach, south of Jacksonville are found from Figures 
5.4 and 5.5. The azimuth angle of the perpendicular to the 
shoreline is 76°-30' as shown in Figure 5.4. Thus, the 
total Southward drift is 1600 cubic yards per day, and the 
total Northward drift is 810 cubic yards per day from Figure 
5.5. The net littoral drift is 790 cubic yards per day or 
288,000 cubic yards per year to the South. 

Limitations in the simple procedure for calculating 
drift values in the above manner will be discussed, taking 
into account some of the data limitations and data bias. 



176 




Figure 5.4. Azimuth of Normal to Shoreline at 
Ponte Vedra Beach, Florida 



177 



Annudlly Av*roij9d Totol Littoral Onff 

St. John's Rivor to St. Augusiltie lnl«t 

N 




Figure 5.5. Determination of Total Positive and Total 

Negative Littoral Sand Transport 



178 



5.2*2 Possible Sources of -Data EvjpQr .Qr ^a s 

In the SSMO data, possible sources of error include: 

(1) Human error and bias in the observation and 
recording of the wave data. 

(2) Absence of extreme wave conditions due to 
routing of ships out of bad weather. 

(3) Inaccuracies introduced due to the lack of 
swell direction data, 

(4) Inadequate resolution of wave data direction wise. 

(5) Inaccurate wave height recording due to wave 
observation in a strong ocean current. 

Error sources (1) and (2) are self-explanatory. In 
regard to (1), it has been shown that a large bias is intro- 
duced in the directional data due to the observer tendency 
toward recording of wave directions along the four cardinal 
and four intercardinal .points of the compass. This effect 
is seen in the littoral drift roses. It is felt that the 
bias should not significantly affect the results presented 
here though, since wave directions used in the- computations 
were reduced to the eight points of the compass in the SSMO 
volumes. If it is assumed that the waves were recorded to 
the nearest point of the compass (on an eight point system) , 
the maximum error between a recorded wave direction and its 
true direction would be 22-1/2°. It is recommended that 
values of drift in a range of azimuth angles 11-1/4-° to 



179 



i:he actual coastlin-e aziinu-th- i>e ■Tso-nsadeT-ed as the range of 
possible drift values, thus covering a 22-1/2° range of 
possible wave directional error. 

The original method of reducing the data from 36 points 
of the compass to 8 points of the compass given in the SSMO 
volumes introduced a skew of the data by an angle of 10 
degrees clockwise. This has been compensated for in the 
littoral drift roses and offshore wave climate roses by 
shifting rose azimuth angles 10 degrees counterclockwise. 

As mentioned earlier, the lack of swell direction data, 
and distinction between sea and swell, cause the assump- 
tion to be made that swell waves are being propagated in 
the same direction as the local wind waves (which is the 
recorded wind direction) . It is unlikely that swell is 
always in the same direction as the local seas and this 
could lead to considerable error in the computation of long- 
shore sand transport. In regard to Item ("4), since long- 
shore sand transport is dependent on wave direction due 
to refraction process, the method of computing sand trans- 
port by using only eight points of the compass poses a 
question as to the magnitude of error possible in the 
results. It can be shown that the maximum error introduced 
by this approach as compared to spreading the energy 
evenly over all directions within an octant is 10 percent. 



180 



In regard to Item .( 5) ,.-.wave- iaights are affected by 
strong currents and have the tendency to steepen when 
propagating against an opposing current and are reduced 
in height by a following current. This effect is noted on 
the Southeast coast of Florida where the Gulf Stream is 
very close to shore. Due to the fact that shipping lanes 
run through and along the Gulf Stream, it is felt that many 
of the ob&erved waves approaching shore have recorded 
wave heights higher or lower than would be experienced on 
the shoreward side of the Gulf Stream in comparitively 
still water. This effect would cause the computed South- 
ward drift values to be higher than the actual drift values 
and Northward drift values to be lower. 

5.2.3 Other Possible Errors or Bias in Analysis 

A possible large source of error comes from the 
assumption mentioned earlier that waves are considered to 
be propagating in one direction at a time. That is, it is 
assumed that when waves are moving away from the coast, 
there are no waves reaching the coast, and thus there is no 
sand transport at the coast. This is a questionable assump- 
tion since waves are known to propagate in many directions 
at the same time but analysis of such error is impossible. 

Other possible sources of error involve the computation 
of the friction coefficient and the violation of Snell's 
Law with regard to the bottom contours . 



181 



.The modification of wave height due. to friction effects 
as the wave propagates across the continental shelf has 
factors which could contribute to inaccuracies as follows; 

(1) Friction coefficient 

(2) Method of taking profile for a coastline section 

(3) Neglecting friction beyond a certain depth, = h 

The friction coefficient used in this study was constant, 
■equal to 0.02, but is known to be a function of bottom 
roughness (as noted in Chapter II), which in turn depends on 
wave height, and water depth. Thus, friction is not constant 
but varies with time. A sensitivity test was done using 
three friction factors: 0.01, 0.02, and 0.0 3 for the location 
of coastline which best represents an average profile from 
Fort Pierce Inlet to St. Lucie Inlet to compute values of 
drift. Assuming that the friction factor 0.02 is correct, a 
value of f=.03 gives drift values approximately 20% lower 
and a value of f=.01 gives drift values approximately 22% 
higher. The sensitivity would be much greater on a broader 
shelf width as in North Florida on the East Coast, and much 
smaller on a narrow shelf width as encountered in the south- 
ern limits of Florida on the East Coast. 

The method of taking a profile perpendicular to the 
stretch of shoreline considered leads to high values 
which would tend to overestimate the wave height and long- 
shore sand transport since refraction effectively causes 



182 



waves to travel over a longer profile than the one used. 
In view of the fact that locations of the individual wave 
data observations are unknown, the method of using a profile 
along the perpendicular to shore seems a reasonable 
approximation though. 

Assumption (3) which was mentioned earlier is made 
because the value of is based on linear wave theory, 
and,, near breaking conditions in shallow water, the wave 
form no longer corresponds to linear theory. In addition, 
the beach is in a dynamic state at shallow depths which 
would make assumptions regarding bottom profile in this 
region invalid during part of the year. 

5.2.4- Other Potential Sources of Error 

Other factors which certainly have a bearing on 
littoral drift in an area but which were not accounted 
for in the present computations include : 

(1) Wind, tidal, and inlet refraction effects on 
littoral currents and corresponding drift. 
■ (2) Sheltering effects of reefs, rock outcroppings , 
large submerged sand ridges, etc. 
(3) Interference in littoral regime due to jetties, 

inlets, rivers, sand sources, sand sinks, etc. 
Factor (1) has been found to be of major significance 
in some studies (12'4). 



183 



+J 
m 
Q 

o 

S 
CO 
oi 

bO 
C 
•H 

m 

t3 

(U 
■P 

3 

a 
e 
o 
o 



M 
(U 

dj 
C 

•H 

bO 
C 

w 

o 

CO 
a, 
u 
o 
u 

>, 

<U 
-M 

m 
e 

•H 

■P 

CO 

W 



o 

•H 

O 
0} 

u 



(U CO 



ra 

> 



10 
3 
C 

c 
< 



u 
m 

o 

•H 

3 
U 

C 



x: x: 4= j::; x: .c .n xz - x: 
■p-p-p+jp+j-p-p+j-p-p 

ooooooooooo 

CQCOCOWCOWWC/DCOUDCO 































CO 


























o 
























to 


( — 1 


C iH 


iH 


jH 




(J> 


CN 




H 




CO 


CN 






tO CN 


LO 


C30 




H 


d" 




LO 


J" 


rH 


cn 






CO 


CO 


< — 1 


CM 


rH 


r-\ 




CO 


CO 


CO 


CO 


+-' 




1 


t 


1 


1 


1 


1 


1 


1 


1 


1 


1 


4h 


H 


0 f~ 




J" 


rH 








LO 


H 


CO 


CN 




rd 


A iH 






O 


cn 




CO 


CO 


CD 


CO 


CO 






\ CN 


iH 




CM 


CO 




H 


CM 


CN 




CO 


(—4 




>H 












* — ' 








- 


-H 


U 


iH 


















































cu 
























Q) 


w 
























tiO 


























rd 




















































*D 








C*^ 




CO 


o 








































CJ 




CO 


CO 


r— 1 


CN 


CO 


IT 


CO 


LO 


cn 


CO 




•H 


LO 


cn 




J" 


H 


LO 




CO 




LO 


CO 






CM 


CN 




CM 








CO 


CO 


CN 


CO 






























O 
























c 




















































<I 


■H 




















































O 


























■H 


^ 


x^ 






^ 






sz 


^ 


^ 






-P 




-p 


-p 


■p 


P 


+j 


+j 


4-' 




■P 


■p 




O 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 






O 


O 


o 


o 


O 


O 


O 


O 


O 


O 


o 






CO 


CO 


CO 


c/: 


00 


CO 


CO 


CO 


C/3 


C/0 


CO 




*H 


























Q 
























0) 


CO 
























4J 


o 
























n3 






















































+ 
























-P 




























u 
























-H 




























0) 
























Q 






























O 


o 


o 


o 




o 


O 


o 


O 


O 


o 


OJ 


0) 


o 


o 


o 


LO 




LO 


CO 


CO 


CM 


LO 


H 


is: 


a, 


LT) 


1 


Ln 


CO 


1 
1 


CN 
1 


CM 


CM 


H 







o 
o 



o 
o 

CM 



+j -p 

0! 0) 











H 






+J 














C 


C 


u 




0) 




+J 








0) 


M 


1— 1 


o 






P" 


(U 


P> 




P" 


> 








cu 


c 




H 


0) 


c 


(a 




0) 


c 




H 


1— 1 




C 


r-H 


o 




OS 


c 


o 




C 




C 


M 


G 


•H 


o 




•H 






M 




1— 1 




1— 1 


P) 


o 


CO 


■P 








o 




xi 




K, 






CO 




H 


C 




CU 




O 


O 


o 


c 


3 


0) 


rd 


OJ 


<u 


•H 


u 


U 


O 


•H 






!^ 


■H 


•H 


O 


o 


O 


hJ 


■P 


o 


3 




0) 


P> 




3 




^ 




C3 




< 


(U 


> 


M 








CO 




03 






o 


(T3 


03 


+-> 




(1) 


H 




■H 






c 


G 


.a 








rH 




-P 


+j 




o 


03 


0) 


o 


P 


03 


•H 




< 


00 


CO 


&- 


O 


CO 




CO 




a: 



0) 

o 

ill 
P" 

CO C 
<a w 

(0 ?H 

iH O 

u u 

0) 03 

> x 
w 

P" 

O 
CU 



e 

03 



J3 

3 

o 

CO 



4= ^ 



P" P" 

3 3 

O O 



CO CO 00 CO CO 



LO CM LO CO CO o cn 

CO CO CD H cn 00 o 

cn LO o CO o cn 

H H 

I I I I I I I 

LO O r-l '"^ LO 

• r- rH . 

J- CO w . . CO 

CO LO z^ CO LO 

:i CO 



CN cn cn LO CO ^ :^ 



rH rH CD 
r~- CM CO 



P" P" 

3 

O O 

CO s 



PJ pl 

3 3 

O O 

CO CO 



O CO CO 

CO LO r- 
I I 
o 

CN 
O 



J3 ^ ^ 

¥> V ^ 

fn 3 3 

O O O 

S: CO CO 



CD CN O O 
CO CM J- 



O O 

CO rH 



















pj 










G 






CO 




SZ 






(0 


T3 


CO 


03 




O 






rH 


C 


CO 


O 










CO 


03 


03 


O 


CO 


Be 


et 




l-H 


Si 




CPl 


CO 




rH 




03 


1— 1 


u 


rH 


03 


CO 


C 




■H 




0) 


3 


Ph 


U 


1-H 


CO 


SU 


0) 


P 


CD 




0) 




cn 


03 




03 




G 


>, 


0) 


03 


S 


3 


:s 


U 


O 


s 


O 


P-. 




CO 


u 


0) 


X) 




•H 




03 


03 


m 




fn 




G 




C 


CP 


CD 


O 


O 


p 






C 


fn 


rH 




CD 




> 


2: 


< 


&-1 





184 



+j +j +-> 

CO W M 

Q) OJ 0) 

3: S 13 



03 



o 

CM O 

C-l CO CO 

I I I 

^ :^ in 

CO CO C7> 

CN CM iH 



^ CN 
I> CD LO 
04 CN CM 



■P +J -P 

CO CO w 

0) (U (d 

S S W 



c 
o 

•H 

•p 
o 

OJ 
•H 

Q) 
'X3 

O 

o 

o 

<u 
CO 
c 
<u 

CO 

•p 

•H 

CO 

o 
&. 
a. 
o 

<u 
x: 



IT) ITS O 
ID CD ID 
H 



0) 





V — 








-3 






■p 








CO 


< 






to 




CO 




o 




CO 






CO 


rd 






CO 


a. 




Ip 


03 






H 


cu 


rO 


CO 


a 




H 


to 


a 


o 


O 


[0 






O 


Oi 




•H 


to 




0} 




CO 


■p 






C 


m 




4) 


0) 


to 




Oi 




w 



4m 
•P 

to 

0) 
+J 

03 

O 
•P 
T3 

C 



CO 

CO 

■P 

0) 
03 

P4 



185 



The sheltering effect of reefs and rock outcroppings is 
certainly a factor affecting littoral drift along the south- 
east coast of Florida. Many rock outcroppings and reefs 
exist in the littoral regime and definitely influence drift 
values. In places such as Cape Kennedy where a large 
underwater sand ridge exists, the drift pattern is altered 
by the sheltering effect of the ridge which prevents some 
northeasterly wav^. from -reaching the .southern shore and 
some southeasterly waves from reaching the northern shore; 
thus, to an extent, the ridge tends to be a self-perpetuating 
littoral barrier. 

Jetties, inlets, rivers, submarine valley, etc. all 
influence the pattern of drift to alter it from the idealized 
model used to compute values of drift; these influences must 
be recognized when applying drift values derived by the 
approach presented. 

5 . 3 Comparison of Calculated' Littoral' Drift Rates 
W3.th Previously Estimated Values 

Comparisons of the present study results with estimated 
values of net drift compiled by the U.S. Army Corps of 
Engineers are summarized in Table 5.1. The Corps of Engi- 
neers values were determined by various methods which include 
analysis of dredging records, volumetric surveys, and pumping 
records at existing by-pass plants. Computed values of 
drift by the present method provide both an "expected" value 



186 



of drift and, to illustrate the sensitivity of drift to 
coastline orientation, a range of drift values which encom- 
pass +11 1/4-° span of azimuths to the actual coastline 
azimuth, 8^, at a given location. Total positive and nega- 
tive drift rates with corresponding ranges of values are 
summarized in Table 5.2. 

Note that some of the comparisons presented in Table 
5.1 and Table 5.2 may be misleading in that they are ex- 
tremely close to Corps estimated values when the assumptions 
involved in the program to compute drift are possibly vio- 
lated. One such location is Fort Myers Beach on the Gulf 
Coast. The computed value of net drift is 21,900 cubic 
yards per year in a Northerly direction which is extremely 
close to the Corps estimated value of 22,000 cubic yards per 
year. The assumption of parallel offshore contours is 
violated here though, and, refraction of waves from a North- 
easterly direction is undoubtedly much different from what 
the simplified analysis based on Snells ' Law would compute 
it to be. Refraction of N W Waves off Sanibel Island would 
tend to create a complex nearshore current situation with 
the probable direction of drift being North even if the wave 
climate and ideal bottom topography would normally tend to 
create a Southerly drift. 

It is suggested that use of the results presented in 
the littoral drift roses be carried out with a knowledge of 



187 



• 


cn 


CO 




• 


bo 




G 




■H 


(U 


CO 






■H 








>) 


0) 




■P 










0) 


s 


■P 


o 


nJ 




B 




•H 




+J ■ 




W 




H 




CO 




(0 




CO 




(U 




+-> 




to 








■P • 










ca 




U 


a +j 


<u 


CO 






c 


to -P 


•H 


fn C 




O d) 




•P CO 


w 


+J 0) 




•H U 




J &. 


o 


H 0) 


w 


to x: 




■p +J 


?^ 


o 


0 




o 


•H 








bO Id 




to 0) 






T3 


OJ lO 


<U 


> rH 


■P 


< 3 


to 


o 


s 


H H 


•H 


to to 


■P 


3 a 


CO 


c 


w 


C tn 




< to 








O C 




to 




c 




O CQ 








•H 0) 




U 0) 




to c 








S M 




0 c 




a w 





to 

(0 

o 

o 







• 


O 








H 


H 


,- — s 


a 












0) 




c 








•H 


• 




Q) 








+J 


CD 




o 


CO 


+-> 






to 


to 






W 


Pi 




H 








to 




+J 




+-> 


o 






o 




•H 


CO 


El 






T3 






O 





0) H 



in 
<U 
■P 
CO 



H 
10 

+J O 
O w 



+J 0) 



CO 

• o 

O 
C 

•H • 

0) >, 

to 

4h • 
•H CO 



CO 

• o 

a 



(7D ro 

w 

to 
■p 



u 
o 

o 



tU >i 

■p 

to u 

■P 

Hh . 

•H CO 

O >- 



f4 >1 



<D 
+-I 
O 



It u 
■P o 
O w 
H 



CO 

• o 

CD 

c 

•H • 

CD >, 
■P 

to U 



■p 

•H CO 

U 



C 

o 

•H 

4-> 

to 
o 

o 



0) 

c 

to 

o 
J- 



0) 
C 

to 

o 
J- 



o 


CM 


cn 


in 


H 






CN 




CD 


H 


cn 


r- 


00 




<y> 


CO 




CD 


CD 


(J) 




J- 


try 




ID' TO 




lO 
1 


in 


=i- 

1 


IT) 


1 

t>- 


1 

UD 




1 

CD 


1 

to 


1 

o 


1 

OD 


H 


1 

H 


CD 


1 

CO 




cn 




H 


J- 


CD 


(X) 




H 


CD 


CO 




J- 


m 




CN 


CM 








J- 


in 



CT) CO CD CO in 
r~ CD CD CO CO 

J- in CO J- CN 





r- 


cn 


CO 


CO 


cn 




cn 






cn 




CO 


CO 


in 


in 




in 



o 




o 


tn 


in 


rH 


J- 


CD 


O 


CO 


.H 


in 


o 


CD 


H 


cn 


ca 


o 


t> 


in 


CSI 


O 


CN 


CO 


CO 


CN 


04 


CN 


CO 

1 


CN 


CN 


^ 


CM 


1 

CJl 


1 

H 


1 


1 

CO 


1 

CN 


1 

in 


c-^ 


1 

H 


1 

o 


1 

CO 


1 

cn 


CD 


CN 


o 


CD 


i> 


=s- 


CO 


H 


C71 


CO 


i> 




CN 


CO 


r-i 


CN 


CM 


CM 


CN 









CM o cn 
CN o 

CN CN CO 



u 

0) 

> 

Pi 



o 



c 
o 



■p -p 
CO CO 



tu 
c 

•H 

-P 

CO 

3 0) 
bO 13 

o 
c 
o 
p4 



CN 

cn 

M CM 



o o o 
o o o 

CO CD CD 



o o o 
o o o 
H H H 



!h 

O -P 

^ tu 

to c 

K M 

H G 

lO to 

!h -H 

(1) -P 

> CO 

rO to 

G ^ 

tO <U 

(l; CO 



•P 

0) 
C 



<I) 



c 

O 



^^ • 

O -P 

cn 



r-i 

c 

-p 
o 

lO 



CU 

o 
G 
to 
u 
-p 

CO G 
0) w 
T3 

to 

r-l O 

W) ^ 

?^ u 

CD ro 
> K 



o 


• 


CO 


cn 


CO 


cn 


CN 


CO 


CN 


CD 


CO 


CD 


CO 




■H 


to 

1 


rH-rH-H 

1 1 1 


H 
1 


H 
1 


1 

CN 


CO 




r- 


cn 


CO 


in 


o 




CO 


o 








H 


CO 


1 — i 


1 






CM 




J- 






CD 




CO 














H 










CO 


H 












in 


CO 








in 




rH 


iH 


in 






• 


- 


1 


1 




in 


CO 




O 


o 


^ 


CM 




in 


in 


OJ 




CO 


00 






H 


H 


rH 






in 




cn 








r- 




CN 






cn 


H 




r- 


o 


in 


rH 


CO 


CO 


CD 




rH 






H 


H 


c- 


1 

in 


1 

H 


1 

in 


1 

o 


1 

in 


i 

t> 


1 

rH 


CD 


CN 


CD 




CO 


in 


H 


CO 


CD 


in 




in 


(D 


m 



CO CO CD iH 



rH 


CO 


CO 


CO 




CD 


CO 


O 


CO 


CO 


o 


zt 


CO 


O 


rH 




co 


CO 




00 


CD 


cn 


H 


CN 


CM 


CO 


CN 




CN 


H 










rH 


rH 



I I 

CO CO 



p to 
O -H 



+-> 

CO 

to 
o 
o 

Mh 

rH 

c:3 

0) 

o 



o 
to 

CD 

to pq 
CO 
n3 
C-i 



■P 

CD 



CO G 
U tH 
(U 

>, <D 



T3 
G 

to 

rH 

CO 



10 
•H 



C 

o 

U • G 

O -P 0) 

C9 > 



CO to 
to S 

Pi 



a < 



T3 CO 

C CO 

to to 

H Ph 

m 

p to 

CQ U 

lO IB 

<U dJ 
rH 

H CJ 



188 



J- c- to 

CN CM CM 



o cn cN 

H O O 
iH H r-H 



cn m c~~ 

H CM o 
H H H 



H CD CD 

O) OJ o 

CO jj- 

I I I 

03 iH H 

in ID CN 

CO CO CO 



rH 

CD H U3 
CO ^ CO 



(0 



+-> 








w 


< 






m 




M 




0 




M 




o 




n3 






M 






M-i 


to 








CU 


rd 


to 








M 


o 


0 


O 


rO 




x) 


O 


p.. 




■H 


03 




<U 


T3 


Kl 


■M 






C 


CO 


a, 


0) 


OJ 


03 


3 




a< 


W 



189 



the assumptions present in the'study such that one is 
not misled by the seemingly good comparisons as given 
above which may be fortuitous . 

Along the Atlantic Coast, the SSMO data confirm the 
Corps estimates of net Southerly movement of sand, and 
on most of the lower Gulf Coast the data confirm net sand 
movements in a Southerly direction. Along the Panhandle 
the net movement is in a Westerly direction which also 
agrees with other studies. Except for certain anom.alies 
in drift directions due to coastline orientation, the over- 
all trends confirm past observations with regard to direction, 
although magnitudes are different. Reason for the extremely 
high values of drift computed in Southeast Florida are not 
known at this time although the author speculates three 
possibilities : 

(1) The effect of the Gulf Stream current 

on wave height observations as mentioned 
earlier. 

(2) Effects of the Bahama Banks. 

(3) Resolution of wave data into large data 
squares rather than smaller squares where 
overall offshore conditions are the same. 

An interesting observation was made in this study with 
regard to null points in net drift. By viewing either the 
net drift diagrams or the total positive and total negative 



190 



drift diagrams, it can be seen that two types of null points 
exists in the drift regime. In Figure 5.6(a) a "Type 1" 
null point is shown for a portion of a typical total drift 
diagram. Assume first that an island exists such that its 
original orientation conforms to the null drift point (total 
positive drift = total negative drift), Figure 5.6(b). A 
perturbation in the system such as a storm, or the building 
of jetties ^at ends xsf.. the. .i^lan.d could cause the sand to be 
shifted to a position shown in Figure 5.6(c). In this case 
the net drift on the right side of the island would now be 
to the right while the net drift on the left side of the 
island would be to the left. Thus the overall effect of the 
perturbation would produce instability in the island, with 
the net result that the perturbation would increase and 
eventually the island would experience a breakthrough as 
shown in Figure 5.6. 

The orientation of the Gulf Coast shoreline in Lee 
County, Florida, is found to be approximately characterized 
by Type 1 null point. It is noted that this section of 
coastline contains numerous inlets and has a history of 
inlet breakthroughs. Another area where this type of null 
point is experienced is the Gulf Coast near St. Petersburg. 
Islands in this region tend to be extremely concave and 
would have probably broken through by now if not for the 
extensive groin fields hindering the transport of drift in 



191 




ji"^ (b) Ideal Island oriented 

1o null point -zero 
net drift 





Perturbation in system 
causes orientation of island 
with associated drift pattern 





(d) Instability leads to eventual 
breaitthrough 



Figure 5.6. Ideal Case of an Unstable Null Point 



192 



the region. It should be noted that a perturbation in the 
convex sense would also be unstable and lead to an increasing 
convexity. No cases of this type system were noted on the 
open coast but numerous such features are seen in bays due 
to this type drift system. 

The second type of null point is shown in Figure 5.7(a). 
An ideal island when oriented to this type of null point has 
a tejid-ency to sta b-ilxze... atae.l.f once ^ perturbation in the 
system drives it from the ideal state. Figures 5.7(b), 
5.7(c), and 5.7(d) show the series of events leading to 
stability. Part of the East Coast of Florida is near this 
type of null point where a predominant tendency for few 
inlets exists. Many of the inlets (such as Sebastian Inlet 
which occurs very near a "Type 2" null point) have had a 
record of numerous closures after being cut. Of course, 
many additional factors, influence stability and instability 
in true physical systems such as the amount of drift supplied 
to an area, and the ocean tidal ranges. These additional 
effects may overshadow those discussed here. It is hoped 
that in the future this theory can be explored further. 

Some further insight into the "null point" type can be 
given by considering two different cases of offshore wave 
climate, a unidirectional wave climate (all waves from one 
direction) and a bimodal wave climate (waves coming from 2 
directions, 18 0 degrees separate). For simplification the 



193 




(a)- Type H null point 



3n 




(b) Ideal Island oriented 

to null point — zero 
net drift 




(c5 Perturbation in system couses 
orianlolion of Island witli 
associoted drift pattern 




id) Self-stobiiizing drift pattern 
leads to original isiond 
conditions 



Figure 5.7.' Ideal Case of a Stable Null Point 



194 



sand transport wi 1 1 .all .be .considered to .be of suspended 
load and the sand transport equation to be of the simplified 
form 



= Ep sin2a^ (5.14) 



where 



- — * W( S -1) (1-p "7 constant (5.15) 

g ^o ■ 

Now when waves are coming from a predominant direction as 
shown in Figure 5.8, a littoral drift rose can be postulated 
as shown in Figure 5.8 where it has been assumed for simplic- 
ity that Ep=1.0. As the littoral drift rose drawn in Figure 
5.8 is a "net" littoral drift rose it is apparent that this 
type of drift rose applies to the type II shoreline, a 
stable shoreline. 

When waves are coming from the directions as shown in 
Figure 5.9, and each directional climate contains the same 
amount of energy, a "littoral drift rose" can be postulated 
as shown in Figure 5.9. This drift rose can be seen to be 
of the unstable type of drift rose which leads to pertuba- 
tions in the shoreline. This type of wave climate and drift 
rose is believed to be the cause of the large pertubations 
(like small capes) in the shorelines of many elongated bays. 
An example of such a bay is Santa Rosa Sound in the Panhandle 
of Florida where a number of such anomalous shoreline "capes" 
exist (see Figure 5.10). The types of wave climate expected 



195 



WAVE CLIMATE CONSISTS OF 
ONE DOMINANT WAVE DIRECTION 



i 



Q^-^ ECg sin 20 



^ y> y y y y 




Figure 5.8. 



Stable Type Littoral Drift Rose 
Due to Unimodal Wave Climate 



196 



Q^/v Ecg sin 20 



WAVE CUMATE Bl MODAL 
FftRALLEL TD SHORELINE 
FOR AZMUTH SHOWN 




Figure 5.9. 



Unstable Type Littoral Drift Rose 
Due to Bimodal Type Wave Climate 



197 




198 



in these elongated bodies of water are conducive to "littoral 
drift roses" of the unstable type where a long fetch along 
the axis of the bay or sound allows large wave energy compo- 
nents coming from large angles while little wave energy 
comes from directly across the sound due to a restricted 
fetch. This is the type of wave climate conceptualized in 
Figure 5.9. Previously no explanations for these shoreline 
..shapes existed. 

5 . 4 Comparison of Estimated and Observed Wave Climates 
To determine the reliability of the SSMO data and 
computed shoaling, refraction, etc. effects in the present 
study, a comparison was made using wave records obtained 
from shore-based gages. Data from step resistance wave 
gages operated by the Coastal Engineering Research Center 
were made available for three wave gage stations : Daytona 
Beach (East Coast), Lake Worth- Palm Beach (East Coast), and 
Naples (Gulf Coast). Wave data were obtained intermittently 
during the years of operation of these stations due to 
various storms damaging equipment or structures on which the 
gages were mounted. To avoid a seasonal bias in the shore- 
based recordings, a sample of data best representing the 
average annual conditions was used in each comparison. 
Table 5.3 shows the observation period used, the total 
number of observations, and the depths of these stations. 



199 



In regard to the SSMO data, certain assumptions had to 
be made with respect to the frequency of occurrence for wave 
heights and periods. Only the onshore directed waves were 
used for obvious reasons, which gave an extremely high 
frequency of "calm" conditions (H=0) at shore. It was 
assumed for the plotting of cumulative height curves that 
the sea state at shore is best represented by wave heights 
of JLess, than one -£oot wh.en of f ■shore directed waves were 
being recorded. Most likely, many waves greater than one 
foot would be recorded at shore during this time. This 
assumption gives a poor basis of comparison for recorded and 
observed low wave heights in which the majority of waves 
fall. In the cumulative distribution curves for wave period, 
the assumed frequency of occurrence of a specific wave 
period was assumed equal to the frequency of the on-shore 
directed wave (of a specific period) times one (1.0), divided 
by the total fraction of onshore directed waves . 

Cumulative curves of the plotted wave height and period 
distributions at these three stations are shown in Figures 
5.11 through 5.16. The wave height cumulative curves show 
three sets of points with corresponding smooth curves drawn 
through them, one curve for the CERC gages, one curve for 
the deep water onshore wave climate as recorded by SSMO, and 
one curve for the SSMO wave climate as modified by the 
present study to the depth of the recording wave gage. 



200 



Table 5.3. Recording Periods of Shore Based CERC Wave Gages Used in 
Comparison of Actual to Predicted Shore Wave Climate. 



Daytona Beach, Florida 

(Depth of Wave Gage = 15 ft. MWL) 

February-December 1954 

February-November 1955 

February-March 1956 

January -April 1957 

November-December 1954 



Lake Worth-Palm Beach, Florida 

(Depth of Wave Gage = 18.2 ft. MWL at Lake Worth and 

= 15.7 ft. MWL at Palm Beach) 



January 


-April and 












June 


-December 


1958 


Palm 


Beach 


1161 


observations 


January 


-December 


1960 


Palm 


Beach 


2020 


observations 


January 


-December 


1960 


Palm 


Beach 


1687 


observations 


January 


-April and 












June 


-December 


1961 


Palm 


Beach 


1301 


observations 


January 


-December 


1966 


Lake 


Worth 


1751 


observations 


Naples, 


Florida 
















(Depth of Wave Gage 


= 16.5 ft. 


MWL) 




January 


-December 


1958 






1454 


observations 



1570 observations^* 
1151 observations 
234 observations 
321 observations 
304 observations 



"'Each observation is the significant wave height and period as deter- 
mined from a 7-minute recording of sea surface elevation measured 
using a step resistance type wave gage. 



201 




202 



-a 



o 




i 



203 



as 

•a 

•r- 

o 




CP) 



204 



IT3 

-a 

s- 
o 




0) 

i- 



• 



205 




206 




207 



Since it is assumed that periods are not modified by offshore 
topography, two curves are shown on the wave period cumula- 
tive distribution curves, one for CERC recorded wave climate, 
and one for SSMO recorded wave climate with inherent 
assumptions . 

The curves show that wave heights of the higher energy 
waves are represented well by the modified SSMO data. 
Unfortunately , thojjgh, .p^iods are .poorly defined by the data 
source. Due to the large dependence of wave modification on 
wave periods, it is felt that an even closer correspondence 
to offshore observations might be obtained with improved 
period observations. 



CHAPTER VI 
ANALYTICAL MODELS FOR SHORELINE CHANGE 



6 . 1 Introductory Remarks 

As important as the ability to predict sand transport 
is, the problem of being able to predict shoreline change is 
equal 1 y i mportant when evaJ.iL3,±ixLg .optimal location of build- 
ing in the coastal-beach strip. The effects caused by 
reefs, rock outcroppings along the beach and coastal struc- 
tures (revetment, groins, jetties) on the shoreline are 
important to coastal engineers . A predictive capability in 
determining shoreline changes due to longshore sand trans- 
port changes becomes of primary importance in the design of 
coastal structures also. If accurate predictions cannot be 
made for determining the changes due to coastal structures, 
then such projects may be doomed to failure. 

This chapter first discusses a heuristic model for 
determining an "equilibrium" shoreline (a shoreline along 
which the longshore sand transport is zero) under given 
average wave conditions and is applied to specific areas of 
Florida's coast where nearshore reefs play an important role 
in determinirtg shoreline shape. The model is then given a 
more sound theoretical basis using the longshore sand trans- 
port equation. 



208 



209 



A second model (dynamic and analytical) is then postu- 
lated using the sand transport equation and the equation of 
continuity via linearizing the sand transport equation 
similarly to the work of Pelnard-Considere (125) only with 
different parameters. Analytical solutions to the partial 
differential equation can then be found and solutions for 
various coastal structures (e.g. jetties, beach nourish- 
ments) can.be found and noji-d i Tn e n .siona3J.zed solution graphs 
given. 

6 . 2 Heuristic "Equilibrium" Shoreline Model Development 
It has long been recognized (126, 127) that the shore- 
lines of sandy beach areas protected by headlands follow the 
shape of a logarithmic spiral curve. The most prominent 
logarithmic shoreline features noted in the literature are 
associated with natural rock headlands on what Johnson (128) 
termed as "shorelines of submergence." Shorelines sheltered 
by natural reefs, shoals, and coastal structures can also 
exhibit this logarithmic spiral shape though as shown in 
Figures 6.1, 6.2, and 6.3, exemplifying that such natural 
shapes are not just commonplace on naturally rocky coasts 
but exist on most all shorelines with sedimentary beaches. 
Figure 6.1 represents a large shoreline indentation just 
south of a natural offshore "worm" rock reef on the east 
coast of Florida near Vero Beach where the predominant 
direction of sand transport is southward as shown in Chapter 



Figure 6.1. Longspiral Curve Fit to 

Shoreline at Riomar, 
Florida 



211 




Scale — KilomeJera 



212 



GULF OF 
MEXICO 




Depth Contours in fest. 
Shoreline and Offshore Depth 
Contours from Nautical Chart 
N.O.S. 856 S.C. 



I I L 



0 0.1 Q2 Q3 Q4 05 
Scale— Nautical Miles 



-1 — I I I 



0 .1 2 3 4 .5 
Scale - Kilometers 




Figure 6.2. Logspiral Curve Fit to Shoreline 

at Lacosta Island 



213 




o 
o 

CD 

■o 



fSS 
Q. 



o 



> 

o 



s_ 

•1— 

CL. 

to 

o 



vo 

0) 
S- 



2m 



V. Figure 6.2 represents a large shoreline indentation on 
the lower Gulf Coast of Florida on La Costa Island. This 
section of coast is also believed to have a net southward 
sand transport as per Chapter V. A nearshore shoal appears 
to be the cause of the indentation to the south. Figure 6.3 
shows that this phenomena is not limited to only small 
sections of shoreline but can be somewhat representative of 
a large section of coa^t -such as the Panhandle section of 
Florida west of Cape San Bias where the predominant direc- 
tion of sand transport on the beaches is westward (to the 
west of Panama City). 

As the logarithmic spiral is found to describe smooth 
sections of coast so well it might be reasonable to ask if 
such a fit is just a quirk of nature or whether perhaps 
there is a more justified reason behind the phenomena? Some 
insight can be gained by looking at the equation of the 
logarithmic spiral which is of the form 

R = e^^ (6.1) 

The above equation can be reformulated into an equation of 
the form: 

lnR2 - InR^ = k(e2 - 8^) = kAS (6.2) 

or 

R^/R^ = e^^®. (6.3) 

When A9 is a constant arc size the parameter R2/R-j^ is also 
constant for a given value of "k." By a judicious choice of 



215 

the value "k" and of the use of a given section of the log 
spiral curve, an unlimited number of "smooth" curves can be 
drawn and fit to curved shoreline features. Figures 6.1 
through 5.3 have been plotted using different values of the 
parameter R2 / R-j^ . 

Sea shells also have naturally occurring smooth curves 
and can be found to fit a logarithmic spiral curve quite 
closely also (129).,. There is no physical justification for 
this shape though, and indeed it is fortuitous, as shell 
shape can be found to closely fit more physically reasonable 
shapes such as ideal airfoil curves (129). 

Justification for the log spiral shoreline shape as 
well., appears to be lacking from a physical point of view 
and hence a more physically descriptive system for the 
equilibrium shoreline shape needs to be considered as in the 
following heuristic model. 

The present heuristic model describes the shoreline 
sheltered by a headland (or on a smaller scale, a groin or 
offshore breakwater) using a wave energy diagram consisting 
of representative offshore ship wave height and direction 
observations. Offshore ship data ane summarized by the U.S. 
Naval Weather Service Command in a publication entitled 
"Summary of Synoptic Meteorological Observations (SSMO)" 
(122), as discussed previously and used for computing sand 
transport along the coast. 



216 



To describe the simplest case of the present heuristic 
model. Figure 6.4- is presented with an x-y coordinate system 
parallel and perpendicular to the existing overall trend of 
a large coastline segment. Note that stavting in this 
ahapterj the x-y coovdinate axis convention has been changed 
to aovres-pond to simitar notational convention used by othev 
authors. The y direction now refers to an axis yerpendi- 
oular to the. shorelin& .j^hil& the x axis now refers to the 
axis parallel to the shoreline . In the simplest case of 
equal wave energy flux approaching the coast from all direc- 
tions the wave energy directional distribution is described 
by a semi-circle as in the top portion of Figure 6.4. At a 
point "P" described by coordinates (x, y) in a sheltered 
section of coast, by geometric considerations shown in 
Figure 6.4 it can be reasoned out that the slope of the 
shoreline in plan view, must be described by the equation: 

dy , , $ V 

= tan (2") (5.4) 

if there is to be no longshore component of wave energy flux 
(which has been shown earlier to be proportional to H^'^^) 
striking the shoreline in its final equilibrium shape. It 
should be noted here that the condition of no longshore sand 
transport along the coast is the same condition as that 
generally assumed to give the logarithmic spiral shape 
shoreline by earlier investigators (126, 127). This is a 
static condition since no dynamics of sand transport are 



217 




Figure 5.4. Orientation of a Shoreline Segment 
Having Equal Offshore Wave Energy 



218 



involved in the shoreline shape. Thus only a final shore- 
line shape can be arrived at and not intermediate time 
steps. Upon expanding the equation in terms of $ it can be 
shown that : 

g . tan ( tan-^ y/x ^ ^5^5) 
which can be solved for y in the form 

y^ = Constant • xCl + ^1 + (y/x)^) • (6.6) 

This equation can be solved by iteration for a shoreline 

shape or solved graphically from the original differential 

equation as expressed in Equation 6.4-. 

The more complex case of this simple static equilibrium 

model occurs for the typical case of an unequal distribution 

of wave energy flux impinging on the coast. For example, 

off the east coast of Florida the average wave energy flux 

distribution from, the SSMO data is summarized in Figure 6.5. 

5/2 

This distribution of H is formulated from, tabulated 
visual estimates of ship wave observations taken by commer- 
cial vessels and the military vessels within specified 
latitude longitude grid sections in the previously mentioned 
publication "SSMO." By summarizing the wave energy from 

5/2 

given directions using representative wave energy value H 
multiplied by the frequency of occurrence of that energy 
level, the continuous energy distribution shown can be found 
as a function of a direction angle, "Q." 



219 




Figure 5.5. Wave Energy Rose for Ocean Data Square 

#12 Off East Coast of Florida 



220 



The magnitude of this energy flux level is given by the 

value "E*." As the specific units of measurement for this 

5/2 

wave energy are in (feet) , the given level of energy at a 

specific direction angle can be thought of as an (average 
5/2 

wave height) impinging on the coast from the direction 6 
at any time. This is obviously a simplified assumption to 
the true situation where wave energy is from different 
direritlDxis .at different times o.f the year-^and at different 
energy levels, but this simple representation will suffice 
for the conditions of this model. 

Using the energy distribution so described, the total 
energy striking the coastline in both parallel and perpen- 
dicular directions to the x axis in the sheltered area of 
the coast can be described by the quantities P" and N"^ 
respectively where: 

•180 

v-k'^ r ay 

-cosae 



- J E*'2(e) 

(6.7) 

180 



In the sheltered area of the coast the shoreline will 
take the shape to minimize the longshore energy component of 
the incoming waves, i.e., energy component longshore = 0. 
This condition is shown in Figure 6 . 6 and is represented by 
the equation 



221 




Ei_ = Energy Component Longshore =0 

=-N sinIZ + PcosX2 (in x direction) 



Figure 6.6. 



Energy Diagram of Coastline with 
Equilibrium Coast Shape 



22 2 

longshore en-ergy compoxLent - 0 - -N"sinfi + P^cosfi (6.8) 

or 

w = = ai ^6.9) 

where the quantity P-VN'^ = f(0). Thus the condition of the 
shoreline slope at a point in the sheltered area can be 
ascertained for the given angle by two integrations of 

the offshore wave energy diagram described by equation 6.7. 
It should be noted that this model does not take into consid- 
eration any effects of diffraction and therefore cannot 
model the effects of wave action cutting behind a headland, 
but where a solid structure such as a seawall, jetty, or 
groin is the limiting condition, the model might be expected 
to be very realistic. 

A graphical solution diagram for the east coast of 
Florida where the general shoreline trend is considered to 
run directly north and south is presented in Figure 6.7. At 
vari-ous values of angle $ the slopes have been plotted and 
curves fit to the slope values. The curves so drawn re- 
semble logarithmic spirals quite closely. Looking again at 
the same east coast shoreline area which was shown earlier, 
in Figure 6.1, a curve from Figure 6.7 has been fit to the 
shoreline at Riomar, Florida, in Figure 6.8. As Figure 6.8 
shows, the equilibrium shoreline curve fits quite well with 
proper adjustment of the x-y axis origin. 



223 




Figure 5.7. Solution Curves to Equilibrium Shoreline 
Equation for the East Coast of Florida 



224 




Figure 6.8. Equilibrium Shoreline Shape for Riomar, Florida 



225 



Equilibrium coastline curves for a general shoreline 
tending north-south on the Gulf Coast of Florida using a 
similar Gulf Coast wave energy distribution is shown in 
Figure 6.9. These equilibrium curves also can be approxi- 
mated closely by log spiral curves. Figures 5.10 represents 
the earlier case shown for Lacosta Island, now with the 
equilibrium shoreline orientation given. Here it was found 
jiecessary to shift the .a,:xj s o-f the equilibrium diagrams 4 3- 
48° to get a good shoreline fitting. The necessity of this 
shift to provide a good fitting indicates that a shift in 
the energy distribution diagram is necessary for the "true" 
wave climate, i.e., the predominant wave energy comes more 
from the west than from the northwest of the wave energy 
distribution diagram. 

Figure 6.11 presents an equilibrium shoreline shape for 
the Panhandle Gulf coast of Florida using the wave height 
distribution from data square number 16 of "SSMO." The 
equilibrium shoreline gives an improved fit to the shoreline 
shape in this area over the log spiral curve fit previously 
presented in Figure 6 . 3 provided a shift in the energy 
distribution diagram of 21° is made. This suggests a "true" 
wave energy distribution shifted more toward the west than 
the distribution obtained from the ship wave observations. 

As a final use for this heuristic method of equilibrium 
shoreline shape, coastal structures such as offshore breakwaters 



226 




Figure 6.9. Solution Curves to Equilibrium Shoreline 

for Gulf Coast of Florida 



227 



GULF OF 
MEXICO 



I L 



0 Ql_ 02 Q3 0.4 05 

Scnle— Nautical Miies 
I — I — I — I— ; I I I I 

0 .1 2 .3 4 .5 

Scale — Kilometers 




Shoreline and Offshore Depth 
Contours from Nautical Chart 
N.QS. 845 S.C. 



Figure 6.10. Equilibrium Shoreline Shape 
for Lacosta Island, Florida 



228 




229 



can. be considered to limit the a.va 1 1 able wave energy strik- 
ing a shoreline. The sheltering effect of the breakwaters 
at a point P(x,y) in the lee of the breakwaters is shown in 
Figure 6.12 where the wave energy reaching point P is 
limited between directions <l>^ and Again the equilibrium 

shoreline slopes are described by the equation: 




(6.9) 



where P* and N* are again calculated by integration of the 
wave energy diagram times the cosine and sine of the direc- 
tion angle 8 respectively. Once the slopes have been cal- 
culated, smooth curves through the slopes can provide a set 
of potential shoreline shapes. In the case that the shore- 
line has a long stretch of protected coast both updrift and 
downdrift (i.e., a large number of offshore breakwaters or 
groins are in use) and the potential sand transport is zero 
along the coast, the shoreline shape must reach an equilib- 
rium such that the net accretion seaward of the original 
shoreline is equal to the net erosion shoreward of the 
original shoreline as in Figure 6.12. From the given shape 
postulated, the indentation of the shoreline can be cal- 
culated and thus the effects of building too close to the 
shoreline provided for. 

6 . 3 Analytical Treatment of Shoreline Change Model 
One can also look at the sand transport equation as 
providing a mechanism for determining the "equilibrium 



230 




c 




o 






c 


<D 


o 


w 


u> 


« 

o 


o 

Ik. 


a 


49 






"o 


*S 


a 


a 








w 


a 


a 


II 


II 


a 


<s 



<u 
+-> 

O ra 

+J -r- 

o s- 

• i~ o 

D. I— 



HI 



o 

on 

s- 
o 



+-I 



M- 

O 

+J 
</) 
<T3 
O 

+J 

to 

LJ 
<D 
4J 



03 



o 



CO 

s~ 

CD 
-P 
(C 
3 
•I- ^ 

d) <u 

S- S- 
O CQ 

x: 

ui (U 

S- 

S O 

•r- to 

S- 

XI M- 
•r- O 

^ >1 

cr 

UJ 



s- 



231 



shoreline shape." Once again the sand transport equation is 
of the form 



e, sin2a, e C, cosa, 
D D ^ s D D 



tantj) 2 w 

s 



(6.10) 



-r _ /V- <(l-K)tan3N 

- ^\ f ^ 

w 

with „ 

P. = — H c, sina, cosa, as before which is equivalent 

it 8 b b b 

to the following expression assuming is small and cosa^=1.0 

I, = b!' H, ^''^ sin^a, + s! H, ^^^sina, (6.11) 
5- 1 b bib b 



where 



- (Y ^(l-K)tang s % ,Y jg. 
w ^ 

(5.12) 

e c, 

s 1 

w 

w s 

Now to proceed, it is necessary to linearize the sand 
transport equation in the term sina^ by incorporating some 
average value of sina-j,^ into the above equation, i.e., 

i=M 

(sina, ) = y f. sina, . (6.13) 
b 1 bi 

1 = 1 

where a multidirectional wave climate is now assumed and f^ 

= frequency of occurrence of breaking wave with wave direc- 

tion a, . . Then b" = B"(sina, )" and the sand transport 
bi lib 

equation for a multidirectional wave climate (time averaged) 
is 

M 

I„ = (b; + S*) I f.H.^^^ sin(9 - 6, . ) iZ.l^) 
% 1 1.^^11 s bi 

1 = 1 



- (Y i<(l-K)tane , ^s'^b ,y Jg. 

^1 - ^^w f ^ ^8 



232 



where a, . = 9 - 8, . = breaking wave angle with shoreline 

and 9^^ = defined angle of wave crest at breaking 

6 = defined angle of shoreline 
s ° 

This convention is shown in Figure 6.13. 

Now for the "equilibrium shoreline" shape which has 
been defined as the shape the planform of the shoreline will 
take after sand has shifted sufficiently to provide a condi- 
tion of 1^ = 0 along- all- parls ryf trhe shoreline, the above 
equation ' becomes for a point "P" 

I, = 0 = (B* + S*) y f.H.^''^ since - 9, .) (5.15) 
1 1 1 . 1 a. s bi 

2.-1 

which can be expanded using trigonometric identities and 
solved for the angle .8 to equal 

M 

f.ri. smS, . 

tane = ^ = ^ ^ ^ ^ (6.16) 

s M 

y f .H. ^ cose, . 
. ^, i X bi 
1 = 1 

In the sheltered region of coast the above equation can 

be applied at the boundary with a source of wave data and 

integrated in a step wise fashion with tan 9 = ^ until the 
° ^ s dx 

entire planform equilibrium shoreline shape is described. 
As the first method is more intuitatively clear (therefore 
easier to spot mistakes) the second approach was not needed 
but was described for more theoretical insight into the 
method only. 



233 



y 




ELEVATION VIEW 



Figure 6.13. Definition of Sign Convention for Shoreline Modeling 



234 



Again it is important to note that the method does not 
take into consideration diffraction effects although with a 
very detailed analysis diffraction effects could be con- 
sidered in equation 6.16 by incorporating a diffraction 
coefficient K^^ along with the wave height (e.g., H^K^ 
rather than H^). The diffraction process is discussed in 
Reference (130). 

HJae previx)us " eqiidJLibrduin shoreline" model took into 
account only the static shoreline concept (i.e., where 
boundaries must eventually comply with the condition that 
there is no sand flux across them) . It is apparent that to 
describe the dynamic process of sand transport a time as 
well as spacially dependent model must be derived. 

Using the total sand transport equation as discussed 

previously, one can find the sand transport in terms of 

volumetric rate as before 

K <(1-K) tanB r sin2a, e c, cosa, ^ 
n - c w , J b b s b b| 

^l' WcS^-1) (l-p^)f ^ * \tan4) 2 ^ / 

§ o w s 

(6 .17) 

which by making the small breaking wave angle approximation 

2 

(cosa^ = 1.0) and linearizing the sin term in the bedload 

M 

portion of the equation as before with f^sina^^ = (sina^)" 

the equation becomes 

Q. = F, (B,S ,p ,f ) (— ^ (sina,)'-- + -2-5-1 P. (6.18) 
1 ' g^o' w [tantp b w J £ 



235 



The above sand transport equation can be coupled with 
the longshore sand transport continuity equation 

|X=0 (6.19) 

which states that the difference in sand flux into a control 
volume is balanced by the storage (accretion or erosion) of 
sand in the control volume. The coordinate axis chosen in 
the present situation is shown in Figure 6.13, to correspond 
to other type approaches of a similar nature (125) as men- 
tioned previously (again note the axis change from that of 
previous chapters). 

The value h^ represents the entire depth over which the 
profile changes and consists of the berm height above water 
level "hj^^" plus the depth below water to the limits of 
significant longshore sand movement "D^." An implicit 
assumption in the above equation of continuity is that the 
bottom profile is constant in form and moves continuously 
seaward or shoreward due to the volumetric change in sand 
transport caused by a differential in longshore sand trans- 
port. This effectively says that there is no loss of sand 
to the control volume due to onshore-offshore sand movements 
but that such fluctuations in beach profiles average them- 
selves out to provide a net zero profile change during the 
period of concern. Also any sand flux transport through the 
boundaries parallel to the shore when averaged over the 



236 



period of concern has a net zero value. Fluxes of sand in 
the onshore-offshore direction could be taken into account 
in the present model by an additional sink-source term for 
gain-loss to the beach profile but as no data exist;; for use 
in such a formulation it will not be included. The problem 
of onshore-offshore sand transport is in need of consider- 
able research which takes into account both theoretical 
aspects of the problejn.as weJ.1 as practical data collection 
but will not be pursued further here. 

Now the sand transport equation can be differentiated 
with the assumption of small breaking angles as before 

sina, ~ a, = 0 - 0, . , and an additional assumption that 
b b s bi' ^ 

other quantities such as beach slope and wave height do not 
change in the longshore direction. Therefore the continuity 
equation becomes 



t 8t 



1 



e, (sina, )" e c, 
b ■ b s b 



tan(j) 



w 



1 



where 



Now with 



= arctan 



3y 



3x 



e, (sina, )* e^c, 
b b _^ s b 



tH. 



and 



^b^b 



tancj) 
2 

b . 



w 



3 y 

3x 
(6.20) 

(6 .21) 



(6.22) 



along with the assumption that (■^)^ ^'^l we find 



at ^h^ 



E <! ) ^ 

Vh^ . 2 
3x 



(6.23) 



237 



setting 




(6.24) 



which is the assumed constant for a given set of wave and 
sediment parameters . 

The equation can then be reduced to the form of the one 
dimensional heat flux equation for which various solutions 
exist (131 ) . 



The value of K* can be referred to as a "shoreline diffusiv- 
ity" constant. 

• The first solution discussed will be that which was 
solved first by Pelnard-Considere (125) under a differently 
postulated sand transport model. 

This situation is that of a coastal structure across 
the surf zone (i.e., groin or jetty) which stops all sand 
transport. For the condition of no sand transport at the 
location x = 0 (the structure location) , the boundary condi- 
tion to be fulfilled is that = 0 , or in other words 

dv ... 

tanG^ = A second boundary condition xs round at x = " 

where y = 0, assuming the initial condition is that y = 0 
for time t <_ 0 . 

The solution for this case (although with a different 
value of K" ) was found by Pelnard-Considere to be 




(6.25) 



238 



y = tanG, j— exp - ( — - — ) — erfc(— — ) 1(6. 26) 

^ ^/^ 2/K^ 2/K^ 2/K^ 

where erfcC ) is the complementar'y error function and is 
tabulated in various mathmatical handbooks (132). 

The solution can be made dimensionless by dividing by 
the length = 2/K*t tan 6^. A solution graph for this spe- 
cific application is presented in Figure 6.14. The value of 
K* can be found • f ro m - equa-tion 6.-2*4 and the -resulting graph 
used to plot solutions of the shoreline at various time 
increments . 

Pelnard-Considere (125) also considered the case in 

which the structure had filled to capacity and bypassing 

began to take place. The time required for the structure to 

fill "t=t^ can be found from the previous solution for the 

ordinate x = 0 where Y = length of structure = 

K^t, 1/2 
2 (— ^) tane, . 

IT D 

Pelnard-Considere (125) provides a second solution for 
times after the structure has filled to capacity. The 
boundary conditions for his second solution are that y = Y at 
X = 0, and y = 0 at x = =° for all times. The initial condi- 
tions are as in the previous solution y = 0 at t = 0 for 
X > 0. The solution to these specific boundary conditions 
are as follows: 

y = Y erfc( — ) (6.27) 

2/K^^ 



239 




240 



which can be made dimensionless by dividing the above equa- 
tion by the length of structure Y. The dimensionless solu- 
tion is presented graphically in Figure 6.15. Pelnard- 
Considere used a time = t in equation 6.2 7 such that areas 
of shoreline above the x axis would be equal at the time 
when the structure is just filled to capacity in equation 
6.25, i.e., matched solution plan areas. In this manner -t^ 
was found to be t2 = t - G.38t^. Although the matched 
solution of Pelnard-Considere was not exact it appears very 
useful for practical application. 

Other applications of the same linearized partial 
differential equation 6.2 5 can be used in prediction of 
beach nourishment fill changes upon suitable evaluation of 
the "shoreline diffusivity" coefficient k" . Some applications 
are as follows: 

— Beach nourishment fill on a straight reach of 

beach. Fill exists from -x <x<+x and extends Y 

a a 

distance from the original beach. The solution 
for this specific case is as follows: 

y = \ {erf ( )(1-^) + erf( )(1 + (6.28) 

I 2yWT a 2/K^ "^a ^ 

The nondimensionalized solution graph for various fill times 

is shown in Figure 6.16. 

— Point source type of beach nourishment such as 
might exist at a location where a truck dumping of 



241 





■■ - 1 1 ■ ■ 1 1 






_l 




1 










\ 






i 
















1 






■ 












1 























1 


1 ■ 













1 




















































- 

- 




















1 




.... .:. 



















































1 












































































. ■ ' 
















1 


























1 










1 






- 






1 


















__ 


























: 1 













■ 


































— 
— 






























































_ 






























, 






























1 
































- 






















[ - - 




_ , 










— — 















































1 ... 1 ■ 11 

I 




















































1 L . 




































— . i—. 




















^.r — ' 




























-z^ — 































"to — 































































































-to — 






























' — £=i 
























— ^ 




































— — 


. . 1 












1 


- 












— ' 




















i — 














































o ■ 




































































.; I.I — .. 










- 






- ■ ■■ — '1 1 

— — — 




































— , 


-n— 


1 


















I — ^_ — 










1 . ! 




1 , _ 


































— 1 
















-es— 




























































1 U- 


— 


r-<r— 








: 






— ~ — 






_j : 




































































-tdr— 














■ 














—J 


n 


























: 























■ 












■ ■ 






1 


-n — 


































r\ 




























— O 






I 



































, — . — _ 


















— H- 


, 

— : _ 

































...ijj., 






















—J 




— < 


, 


CO 


















- 










— — 


'■<_. 








: 


















h— 


1 

















7 




























































































/ 1 




































/ 
































t : — 
















r- 



































































































































— — - 






















































































— . 1 


















































' 












1 












A 








1 

■ 












- ■ - 
















1 — ' 












.... 










































-1 






























-! 




























— 






■ 




















• 
















! 


■ / 






































































































1 






























1 






1 




























































~ 




















t 










r — ■ — 






















— 






) 


■ 






1 


1 . ■ 




1 


















h— 






1 












1 








; 






















































-- 
























































































































































! 








- 






























-\ 
























- 
















\ 1 ■ 










































































■ ! 



































































1 -1 








— 




































_ 
































































— 


— — 


1 










— i— 1 . 













CM 



■Q 

-OJ 



-d 



d 



o 



i. 




O) 




+J 




M- 




< 














OJ 


s_ 


s_ 


=: 




+-> 


4-J 


u 


u 


33 


r3 


i. 


S- 


+-> 


+-> 




00 


+-> 


H- 


fO 


o 








"O 


0) 






•t— 


TD 




OJ 


cn 


-C 




o 




re 




OJ 




a: 


00 








s_ 




o 










■a 






> 




s- 


OO 






o 


0) 








•r- 


o 


^- 


•1 — 




-l-J 












o 




00 





o 



GO 

d 



03 

d 



d 



d 



242 




243 



sand occurs or jdrag scraping of sand from the 
offshore occurs. The solution for this specific 
case (where the original fill has been assumed 
triangular in shape) is: 

Y r 

y = _£ j (l-X)erf (U(l-X))+Cl+X)erf(U(l+X))-2Xerf CUX)+ 



1 , -U^d+X)^^ -U^(l-X)^ o.-(UX) 
(e +e -2e 



(6.29) 



where X 



and U 



(6.30) 



2/Wt 

The nondimensionalized solution graph for various fill 
times is shown in Figure 6.17. 

For the case in which a beach nourishment has been 
placed on an existing beach but a gap has been left in the 
beach nourishment project (such as occurred in a beach 
nourishment project on Jupiter Island, Florida in 19 74), the 
following solution would apply which is similar to eq^uation 
6. 28 . 



y = J \evfc 



— X 

( (1-— ) 

^a 



+ erf c 



( )(!+!-) 
a 



(6 . 31) 



The nondimensional solution graph for this specific 
case is given in Figure 6.18. 

— The case of a semi-infinite beach fill gives the 
shape of the beach fill end and its extent of 



244 




245 




246 




247 



pjTogression down tJne coa^t with time. The solu- 
tion for this specific case is 

y = yjl + -erf i— — )| (6.32) 

The nondimensional solution graph is shown in Figure 6.19. 



CHAPTER VII 
SAND TRANSPORT AiND STORAGE AT INLETS 

As the present sand transport model developed is for 
ideal beaches where gradients in the longshore direction are 
zero, the model would not be expected to work at inlets, 
where in addition to complex bottom topography, the often 
large tidal currents play an important factor in the trans- 
port processes. Little is known about the specifics of 
sand transport at inlets (133) although generalities can 
be dealt with. 

It is known that many inlets act as sand traps by 
storing sand in their outer shoals (133). They become self- 
perpetuating features by their ability to jet sand (which 
comes to the inlets by ' littoral drift) to the outer shoals 
where the sand is either cycled back to the shoreline during 
heavy wave activity or stored permanently in the outer shoal, 
making the shoal larger and consequently causing refraction 
patterns at the inlet conducive to shoal growth. 

In an attempt to determine what the ultimate limits of 
such growth might be, a correlation was made of the sand 
stored in the outer shoals with the size of the inlet as 
determined by its tidal prism P and the amount of wave 
activity acting on its outer shoal as determined by a wave 
climatology parameter dependent on the^ energy flux at the 

248 



249 



inlet. Details of . the -sand voluiae calculatiojis are dis- 
cussed elsewhere (133), and the estimates of tidal prisms 
and wave parameters as follows . 

Tidal prism measurements came from References (134, 
135, 136, 137, and 138). In most all of the cases the tidal 
prism wa;s either measured from current data taken at the 
throat of the inlet or by the "cubature method." Jarrett 
(13 7) discusses the cubature m-ethod in d-etail . In most all 
cases the data when the prism was measured corresponded to 
the survey data from which the estimate of outer bar sand 
volume was made. 

Wave heights and wave periods available were average 

wave heights from wave gages in the nearshore zone (15 to 

20 feet below MLW) from the Coastal Engineering Research 

Center wave gage program. As these wave heights already 

have the measure of continental shelf slope implicit in 

them (energy has been dissipated over the shelf up to the 

wave gage depth) , the basic measure of wave energy used to 

2 7 

separate energy environments was the parameter H T~ (wave 

2 2 
height X wave period ). On mildly/ exposed, moderately 

exposed, and highly exposed coastlines, this parameter was 

arbitrarily chosen to range from 0-30, 30-300, >300 

respectively. This classification lum.ps the South Carolina, 

Texas, and lower Gulf Coast of Florida inlets into the 

mildly exposed coast range; the East Coast, and the Panhandle 

of Florida (Gulf Coast) inlets into the moderately exposed 

coast range; and the Pacific Coast inlets into the highly 



250 



exposed coast range. Cojrjrelations were made for three 
coastal energy level groupings and for all inlets combined 
using an equation: 



^ = aP^ (7.1) 



where 



^ = volume of sand stored in the outer 

bar/ shoal of the inlet (in cubic yards 
of immersed sand) , 

P = tidal prism of inlet (in cubic feet), 
a,b = correlation coefficients. 
Linear logarithmic regression was used to obtain b for the 
case of highly, moderately, and mildly exposed inlets for 
the case of inlets tabulated in (9). 

For these four cases the coefficient b equals: 
Highly Exposed Inlets b = 1.23 

Moderately Exposed Inlets b - 1.08 
iMildly Exposed Inlets b = 1.24 

All Inlets b = 1.26 

As there was no significant difference in the exponential 
correlation coefficients, the value b = 1.2 3 corresponding 
to the high energy coast (Highly exposed) inlets was used 
for the correlations with all inlet groupings. The justifi- 
cation for this somewhat arbitrary fixing of parameters was 
that a minimum of scatter existed in the correlation of the 
Pacific Coast Inlets (Figure 7.1), The minimum scatter in 
this plot over two orders of magnitude is somewhat surprising 
in view of the many parameters which could be of importance 



251 




ure 7.1. Tidal Prism - Outer Bar Storage Relationship 

for Highly Exposed Coasts 



252 



in inlet outer bar shoaling such as inlet history, avail- 
able longshore energy flux, and physiography of the inlet- 
coastal location. 

Some reasons for some of the scatter seen are as 
follows : 

-Inlet history. Should the inlet close 
or the tidal prism be reduced drastically 
(due to m.odif icat ions of the inlet inner 
bay system) much of this material would be 
driven back to the beaches. The author 
has noted this occurrence in two occasions 
on both the East Coast and Gulf Coast of 
Florida. In the case of the Pacific Coast 
inlets studied, all of the inlets have been 
open over recorded history. 

-Physiography must plan an important 
part also. The author has noted that inlets 

i which are estuarine (i.e., estuaries), have 

significantly smaller inner shoals most 
likely due to the predominance of the ebb 
flow during landward flooding periods . The 
Pacific Coast inlets used for the highly 
exposed coast correlation are alike in that 

j they are estuary systems , 



253 



Thus, the Pacific Coast -inlets ajre .physio graphically 
similar, historically open, and have similar (in a gross 
sense) longshore energy flux levels; and, therefore, should 
experience less scatter. 

Using the exponential correlation coefficient b = 1.23, 
analysis was made to determine the correlation coefficient 
a and the corresponding volume-prism relationship for the 
three ■ groupings of inlets and for the inlets combined. 
The corresponding equations are shown below: 

High exposed coasts (7 inlets) 

= 8. 7 X lO'^p^-^^ (7.2) 

Moderately exposed coasts (18 inlets) 

^ = 10 . 5 X 10"'^ (7.3) 

P 

Mildly exposed coasts (15 inlets) 

¥ = 13. 8 X 10~^p^' (7.4) 
All inlets (44 inlets) 

^ = 10.7 X 10~^p^-^^ (7.5) 
The plots of the prism-outer bar storage volume for 
the various inlet groupings are given in Figures 7.1, 7.2, 
7.3, and 7.4. 

As inlet channel cross-sectional area shows a definite 
correlation with tidal prism (134, 135) and is an easier 
quantity to measure than tidal prism, a correlation was also 
made with the available data for bar volume — channel cross- 
sectional area relationships. Correlations were made for 
the three coastal energy level groupings and for all 
inlets combined using an equation: 



254 



a 
rr. 
< 
>- 



1000 - 



500 - 



200 - 



SAN FRANCISCO 9*Y, CALIF. 
MOBILE BAY, ALA . -r/. V_ 
COLUMBIA RIVER, aRP/-~ T" 




NORTH EDISTO RIVER 



< 

03 

X. 

(Jj 
I— 

O 



<5 



'00 -ST. AUGUSTINE INLET, FLA.-f- 

ST. JOHN'S RIVER ENT. FLA.' 



50 - 



GRA/'S HAR30R, WASH. 
+ 

CA SRANOe INLET, FLA. 

ST. Jf'ARY'S RIVER ENTRANCE , FLA. 

GALVESTON , TEXAS 

3TQ<Q0 INLET, 3. C. 

NYAH SAY, S.C. 

SAU SOUND , FLA. 

J- + PENSACQLA HARBOR, FLA. 



/+ 



-aSAU.'^ORT' INLET, N.C. 



+ ^ 

LITTLE EGG INLET, N.J . -p 

COOS 3AY, ORE, , 
ARANSAS PASS, TEXAS/ -i-UMPQUA RIVER, ORE. 

PASS-A-GRILLE INLET, FLA.— ^n-^j. -r OREGON INLET, N.C 

HEREFORD INLET, N.J.-— '"^/" ;+ TILLAMOOK SAY, ORE 

BIG SARASOTA PASS, FLA. 4- y '^GREAT EGG HARBOR INLET, N.J. 
PONCE OE LEON INLET, FLA. 4 



-r CAPTIVA PASS, FLA. 



SARNEGAT INL^T, N.J.-f- 
10 - 

DUNEDIN PASS, FLA. + 4- LONGBOAT PASS, FLA. 
GASPARILLA PASS FLA.-..i.jy^3„,(3^;^j|f^£ INLET, N.J. 
NEW PASS,FLA.+-^ jQ„^'3 PASS, FLA. 

-> -~ - -^J^ -r EAST PASS, FLA. 

+ NEHALEM RIVER, ORE. 
-L>-CAR0L1NA BEACH INLET, N.C. 
INDIAN RIVER INLE^^ QEL.4-+ CLEARWATER INLET, FLA. 

2 - 



REDflSH PASS, rl 



LU 

5 

_J 

O 
> 



•0 ^ JUPITER INLET, FLA. 
+/«ENIC£ INLET, FLA. 

+ MIDNIGHT PASS, FLA. 
0.5 - 



+ SAKER'S HAULOVER , FLA. 

0.2 - 



Q I I 111 III I 

12 5 10 20 50 100 300 SCO lOOO 



TIDAL PRISM IN CUeiC FEET (x 10° 



Figure 7.2. . Tidal Prism - Outer Bar Storage Relationship 

for Inlets on Sandy Coasts 



255 




0.3 ' ' 111 I t I , 

12 5 10 20 50 100 200 500 1000 

TIDAL PRISM IN CUSIC rEET(xlO^) 



Figure 7.3. Tidal Prism - Outer Bar Storage Relationship 

for Moderately Exposed Coasts 



256 




ure 7.4. Tidal Prism - Outer Bar Storage Relationship 

for Mildly Exposed Coasts 



257 



¥ = a'A ^' (7.6) 



c 



where 

= volume of sand stored in outer bar/ shoal 
as before, 

A = inlet channel, cross-section area at 
throat (in square feet), 

a',b' = correlation coefficients 

Linear logarithmic regression was again used to obtain 

the coefficient b' for all cases and is tabulated below: 

Highly Exposed Inlets b' = 1.28 

Moderately Exposed Inlets b' = 1.23 

Mildly Exposed Inlets b' ~ 1.28 

The value b' =1.2 8 corresponding to the highly exposed 

and mildly exposed inlets was used for the correlations 

with all inlet groupings, and analysis was made for the 

coefficient a'. The corresponding equations for all inlet 

groupings are shown below: 

Highly exposed coasts (7 inlets) 

V = 33.1A ^'^^ (7.7) 
c 

Moderately exposed coast (18 inlets) 

1 9 P 

V- = 40 . 7A ^ (7.8) 
c 

Mildly exposed coasts (15 inlets) 

^ = 45. 7A -'^^ (7.9) 
c 

The plots of the cross sectional area-outer bar storage 
volume for the various groupings are not shown, but prove to 
have considerably less scatter than the tidal prism-outer 



258 



bj3.r storage volume plots. Under a given set of conditions, 
hence, both the volume-prism and volume-cross section rela- 
tionships should be considered when obtaining an estimate 
of the sand storage capacity of an outer inlet bar system. 

A nuFiber of parameters other than tidal prism (or 
cross-sectional area) and wave energy also play a large 
role in sand trapping on outer bar/shoals. Two important 
parsm-eter s which have not been -explicitly considered in the 
present analysis are longshore energy flux which moves the 
sand to the inlet where the ebb tidal current can deposit 
on the outer bar, and size distribution of littoral material 
which limits the ability of the material to movement away 
from the surf zone. Further research is needed to better 
define how these parameters control the influence of outer 
bar sand storage. 

Further work is also needed on the inner bay or lagoon 
shoal storage volumes and on the potential of any given 
inlet to trap sand in its interior shoal system. 



CHAPTER VIII 
SUMMARY AND CONCLUSIONS 

In the preceding chapters a model for sand transport 
along beaches under the influence of wave action was devel- 
oped and compared to existing laboratory data and fit to 
within a constant. The model predicts reasonable values for 
the integrated (across the surf zone) littoral bed load and 
suspended load sand transport when compared to existing field 
and laboratory measured sand transport values. As of yet 
-though, instrumentation does not exist to perfectly differenti- 
ate the bed load portion of the sand transport from the sus- 
pended load portion (139), and thus, the individual bed load 
and suspended load sand transport models cannot be verified 
independently . 

As both field and laboratory measurements of sand trans- 
port are difficult to make and very costly, few data sets 
exist which provide all parameters of importance to the cal- 
culation of sand transport and no data have been taken in 
which the distribution of sand transport across the surf zone 
has been measured adequately. Also, field data taken have 
often left out adequate description of the sediment size dis- 
tribution and rely mainly on the mean or median grain size, 
whereas proper measurement should include the fall velocity 
distribution curve (for suspended load) and the grain size 
distribution curve (for bed load) . 



259 



260 



Further research should be done on the following sub- 
jects to improve future sand transport models. 

--Collection of both field and laboratory measure- 
ments of sand transport as well as forcing functions 
(wave parameters) and sediment parameters across the 
surf zone. 

--Collection of data to define lateral "mixing" 
parameter; necessary for predictive ability of dis- 
tribution of longshore current across surf zone. 

— Concentration of suspended sediment in the 
surf zone along with random fluctuations of con- 
centration and velocity to better define the 
vertical sediment mixing parameter. 

--Development of better instrumentation for use 
in the laboratory and field in surf zones, especially 
for measuring instantaneous sediment concentrations . 
Present instrumentation for use in the surf zone is 
not capable of differentiating between entrained air 
bubbles and sediment (139), 

The present model was applied to the prediction of 
sand transport along the Florida Coast in Chapter V using 
existing offshore ship wave data. Computations were made 
by means of computer model which takes into account wave 
height modification due to energy dissipation by bottom 
friction. A com.parison is made to sand transport values 
estimated from dredging records at inlets which provides 
reasonable agreement in most areas. By a means of data 



261 



presentation called the "littoral drift rose" based on 
coastline orientation, the ability to postulate shoreline 
stability and instability is given. Using thei "littoral 
drift rose" concept,:" a rational explanation for the existence 
of natural large cusp features (instability features) oc- 
curring along elongated bays is presented- 

In Chapter VI a heuristic model for the static 
-equilibrium (no longshop.e . sand transport) shape of shorelines 
sheltered by non-erodable materials (headlands, reefs, etc.) 
is presented. The model is compared to existing shoreline 
shapes in the lee of reefs and shoals and is found to 
provide reasonable results inrjnany areas. Using the developed 
equation for sand transport, a dynamic model of shoreline 
change is also postulated. The sand transport model equa- 
tion is coupled with a volumetric continuity (of sand) equa- 
tion and reduced to the "heat flux" form of partial differential 
equations which is then solved for various coastal structure 
(jetty, beach nourishment, etc.) cases. 

Chapter VII presents the results of a regression anal- 
ysis for the volume of sand stored in the outer bars of 
tidal inlets with the results in the form of equations which 
involve both inlet tidal prism and wave activity present 
offshore of the inlet. Results of the analysis allow for 
equilibrium calculation of inlet outer shoal volume in the 
event an inlet is to be cut in a barrier island. 



REFERENCES 



1. U.S. Army Corps of Engineers, National Shoreline Study , 
Washington, D.C., 1971. 

2. Hicks, Steacy D. , "Trends and Variability of Yearly Mean 
Sea Level 1893-1971," NOAA Technical Memo. No. 12 , 
Rockville, Maryland, March 197 3. 

3. Bruun, P., "Sea-Level Rise as a Cause of Shore Erosion," 
Journal of Waterways, Harbors, and Coastal Engineering , 
ASCE, Vol. 88, No. 117, May 1962, pp. 183-217. , 

4. Hayes, Miles 0., "Hurricanes as Geologic Agents: Case 
Studies of Hurricanes Carla, 19 61 and Cindy, 19 6 3," 
Report 61 , Bureau of Economic Geology, University of 
Texas, Austin, Texas, Aug. 196 7. 

5. Chiu, T. Y., and Purpura, J. P., "Study of the Effects 
of Hurricane Eloise on Florida's Beaches," Coastal 
and Oceanographic Engineering Laboratory, Gainesville, 
Florida, Aug. 1977. 

6. Walton, T. L,, Jr., and Dean, R. G., "Use of Outer 

Bars of Inlets as Sources of Beach Nourishment Material," 
Shore and Beach Magazine , Vol. 44, No. 2, July 1976, 
pp. 17-31. 

7. Dean, R. G., and Walton, T. L., "Sediment Transport 
Processes in the Vicinity of Inlets with Special 
Reference to Sand Trapping," Proceedings of the Second 
International Estuarine Research Conference , Vol. 2, 
Myrtle Beach, S.C., Oct. 1973, pp. 129-150. 

8. Walton, T. L. , Jr., "A Relationship Between Inlet Cross 
Section and Outer Bar Storage," Shore and Beach Magazine , 
Vol. 45, No. 2, April 1977, pp. 27-42. 

9. Walton, T. L., Jr., "Capacity of Inlet Outer Bars to 
Store Sand , " Proceedings of the 15th International 
Conference on Coastal Engineering , ASCE, Vol. 2, New 
York, 1976, pp. 1267-1285. 

0. Fisher, J. S., Leatherman, S. P., and Perry, F. C, 

"Overwash Processes on Assateague Island," Proceedings 
of the 14th International Conference on Coastal 
Engineering , ASCE, Vol. 2, New York, 1974, pp. 1211-1229, 



262 



263 



11. Burdin, W. W. , "Surge Effects from Hurricane Eloise," 
• Shore and Beach Magazine , Vol. 45, No. 2, April 1977, 

pp. 1-17. 

12. Purpura, J. A., "Establishment of a. Coastal Setback 
Line in Florida," Proceedings of the 13th International 
Conference on Coastal Engineering , ASCE, Vol. 3, New 
York, 1972, pp. 1789-1809. 

13. Walton, T. L. , Jr., and Purpura, J. A., "Beach Nourish- 
ment Along the Southeast Atlantic and Gulf Coasts," 
Shore and Beach Magazine , Vol. 45, No. 3, July 1977, 
pp. 15-33. 

14. U.S. Army Corps of Engineers, "Investigation of Erosion, 
Carolina Beach Inlet, N.C.," Wilmington District, 
Wilmington, N.C., 1970.' 

15. Information provided by courtesy of Wilriiington District, 
U.S. Army Corps of Engineers, Wilmington, N.C. 

16. Berg, Dennis W. , and Essick, Morrison G., "Case Study, 
Hunting Island Beach, South Carolina," Proceedings of 
the Seminar on Planning and Engineering in the Coastal 
Zone , Charleston, S.C., June 8-9, 1972, pp. 123-139. 

17. Information provided by courtesy of Charleston District, 
U.S. Army Corps of Engineers, Charleston, S.C. 

18. Anonymous, "Report on Monitoring of Beach Fill South 

of Canaveral Jetties, Brevard County, Florida," Coastal 
Engineering Laboratory, University of Florida, 
Gainesville, Fla. , July 1976. 

19. Anonymous, "Study Report to Determine the Behavior of 
Project Fill for Beach Erosion Control, Virginia Key 
and Key Biscayne," Coastal Engineering Laboratory, 
University of Florida, Gainesville, Fla., March 19 72. 

20. Anonymous, "Study Report to Determine the Behavior of 
Project Fill for Beach Erosion Control, Treasure Island, 
Florida," Coastal Engineering Laboratory, University 

of Florida, Gainesville, Fla., Dec. 1971. 

21. Information provided by courtesy of Jacksonville 
District Office, U.S. Army Corps of Engineers, 
Jacksonville, Fla. 

22. Anonymous, "Behavior of Beach Fill and Borrow Area at 
Harrison County, Mississippi," Beach Erosion Board 
Technical Memo No. 10 7, Washington, D.C., Aug. 19 58. 



264 



23. Information provided by courtesy of Mobile District, 
U.S. Army Corps of Engineers, "Mobile, Ala. 

24. Caldwell, J. M, , "Wave Action and Sand Movement Near 
Anaheim Bay, California," Beach Erosion Board Tech - 
nical Memo. No. 68 , Washington, D.C., 19 56, 

25. U.S. Army Corps of Engineers, Shore Protection Manual , 
Washington, D.C., 1973. 

26. Inman, D. L. , and Frauschy, J. D., "Littoral Processes 
and the Development of Shorelines," Proceedings of the 
10th Conference on Coastal Engineering , ASCE, Santa 
Barbara, Cal., 1965, pp. 511-536. 

71. Dean", 'K. G., "Heuristic Models of Sand Transport in 

the Surf Zone," Proceedings of Conference on Engineering 
Dynamics in Surf Zone , Sydney, NSW, 1973, pp. 211-231. 

28. Iwagaki, Y. , and Sawaragi, T., "A New Method for 
Estimation of the Rate of Littoral Sand Drift," Coastal 
Engineering in Japan , Vol. 5, 1969, pp. 67-79. 

29. Le, Mehaute", B,, and Brebner, A., "An Introduction to 
Coastal Morphology and Littoral Processes," Research 
Report No. 14 , Queen's University of Civil Engineering, 
Kingston, Ont. , 1959. 

30. Thornton, E. B., "Variations of Longshore Current 
Across the Surf Zone," Proceedings of the 12th 
Conference on Coastal Engineering , ASCE, Vol. 1, 
Washington, D.C., 1970, pp. 291-308. 

31. Bijker, W. W. , "Longshore Transport Computations," 
Journal of Waterways, Harbors, and Coastal Engineering , 
ASCE, Vol. 97, No. WW4 , Nov. 1971, pp. 687-701. 

32. Savage, R. P., "Laboratory Study of the Effect of 
Groins on the Rate of Littoral Transport: Equipment 
Development and Initial Tests," Beach Erosion Board 
Technical Memo. No. 114 , Washington, D.C., 19 59. 

33. Ijima, T., Sata, S., Aono , H., and Ishii, K. , "An 
Investigation of Sand Transport on the Fukue Coast," 
Coastal Engineering in Japan, Vol. 2, 1960, pp. 59-79. 

34. Ichikawa, T., Ochiai, 0., and Tomita, K., "An Investi- 
gation of Sand Transport," Coastal Engineering in Japan , 
Vol. 33, 1961, pp. 161-167. 



265 



35. Ijima, T. , Sato,.S,, and Tanaka, JJ- , "An Investigation 
of Sand Transport on the Kashima Coast," Coastal 
Engineering in Japan , Vol. 6, 196H, pp. 175-180. 

36. Sato, S., "Sand Transport on the Kashima Coast," 
Coastal Engineering in Japan , Vol. 8, 1966, pp. 19-29. 

37. U.S. Army Corps of Engineers, "Shore Protection, 
Planning and Design," Technical Report Number M- , 
Washington, D.C., 1966. 

38. Komar, P. D. , "The Longshore Transport of Sand on 
Beaches," thesis presented to the University of 
California, at San Diego, Cal., in 1969, in partial 
fulf illmen-t- of the -T^eq-ai-rements- f or the degree of 
Doctor of Philosophy. 

39 . American Society of Civil Engineers , Sedimentation 
Engineering Manual , New York, 1976. 

■40. Inman, D. L. , and Bagnold, R. A., "Littoral Process," 
The Sea , Vol. 3, Interscience Publishers, New York, 
1963, pp. 529-549. 

41. Longuet-Higgins , M. S., "Recent Progress on the Study 
of Longshore Currents," Waves on Beaches , Academic 
Press, New York, 1972, pp. 203-248. 

42. Inman, D. L., "Review of Existing Sediment Transport 
Theories," Proceedings of the National Sediment 
Transport Study Conference , Newark, Del., Oct. 19 76, 
pp. 5-18. 

43. Saville, T., Jr., "Model Study of Sand Transport Along 
an Infinitely Long Straight Beach," Transactions of 
the American Geophysical Union , Vol. 3, No. 4, 1950, 
pp. 555-555. 

44. Putnam, J. A., Munk , W. H. , and Traylor, M. A., "The 
Prediction of Longshore Currents," Transactions of the 
American Geophysical Union , Vol. 30, No. 3, 1949, 

pp. 337-345. 

45. Bagnold, R. A., "An Approach to the Sediment Transport 
Problem from General Physics," U.S.G.S. Professional 
Paper 422-1, Washington, D.C., 1966. 

46. Longuet-Higgins, M.S., "Longshore Currents Generated by 
Obliquely Incident Sea Waves," Part 1 and 2, Journal of 
Geophysics Research , Vol. 73, 1970, pp. 6778-68-1. 



266 



4 7. Bowen, A. H. , "TJie Gejiera±ion of Longshore Currents 
on a Plane Beach," Journal of Marine Research , Vol. 
27, 1969, pp. 206-215. 

48. Brebner, A- , and Kamphuis , J. W., "Model Tests on 
Relationship Between Deep-Water Wave Characteristics and 
Longshore Currents," Research Report 31 , Queen's 
University of Civil Engineering, Kingston, Ont., 1963. 

49. Sonu, C. J., McCloy, J. M. , and McArthur, D. S., 
"Longshore Currents Nearshore Topographies," Proceedings 
of the 10th Conference on Coastal Engineering , ASCE, 
Vol. 2, Tokyo, Japan, 1966, pp. 525-549. 

50. Harrison, W. , and Krusib^in, W. C, , "Interactions of 
the Beach-Ocean-Atmosphere System at Virginia Beach, 
Virginia," Coastal Engineering Research Center 
Technical Memo. No. 7 , Washington, D.C.,1964. 

51. Harrison, W. , "Empirical Equation for Longshore Current 
Velocity," Journal of Geophysics Research , Vol. 73, No. 
22, 1968, pp. 6929-6936. 

52. Inman, D. L., and Quinn, Q. H., "Currents in the Surf 
Zone," Proceedings of the 2nd Conference on Coastal 
Engineering , Gainesville, Florida, 1952, pp. 24-36. 

53. Chiu, T. Y. , and Bruun, P., "Computation of Longshore 
Currents , " Proceedings of the International Conference 
on Coastal Engineering , Lisbon, Portugal, June 19 64, 
p. 197. 

54. Galvin, C. J., and Eagleson, P. A., "Experimental Study 
of Longshore Currents on a Plane Beach," Coastal 
Engineering Research Center Technical Memo. No. 10 , 
Washington, D.C., 1965. 

55. Eagleson, P. S., "Theoretical Study of Longshore 
Currents on a Plane Beach," M.I.T. Hydrodynamics 
Laboratory Technical Report 82 , Cambridge, Mass., 1965. 

56. Galvin, C. J., "Longshore Current Velocity: A Review 
of Theory and Data," Reviews of Geophysics , Vol. 5, No. 
3, 1967, pp. 287-303. 

57. Shepard, F. P., and Sayner, D. B., "Longshore and 
Coastal Currents at Scripps Institution Pier," Beach 
Erosion Board Bulletin , Vol. 7, Washington, D.C., 
1953, pp. 1-9. 



267 



.58- . Putnam, J. A., axid JqIuisidji, . J, W. , "Tiie Dissipation 
of Wave Energy by Bottom Friction," Transactions of 
the American Geophysical Union , Vol. 30, No. 1, 1949, 
pp. 67-74. 

59. Bretschneider , C. L-, and Reid, R. 0 ., "Modification of 
Wave Height Due to Bottom Friction, Percolation, and 
Refraction," Beach Erosion Board Technical Memo. No. 
M-5 , Washington, D.C., 1954. 

60. Bretschneider, C. L. , "Field Investigation of Wave 
Energy Loss of Shallow Water Ocean Waves," Beach 
Erosion Board Technical Memo. No. 46 , Washington, D.C., 
1954. 

61. Iwagaki, Y., and Kakinuma, T. , "On the Bottom Friction 
Factor of the Akita Coast," Coastal Engineering in 
Japan , Vol. 6, 1963, pp. 194-210. 

62. Iwagaki, Y., and Kakinuma, T., "On the Bottom Friction 
Factors off Five Japanese Coasts," Coastal En g ineering 
in Japan , Vol. 10, 1967, pp. 45-53. 

63. Kishi, referred to in Computer Prediction of Nearshore 
Wave Statistics , ONR Contract Report N00014-69-C-010 7 
by Sonu, C, September, 1975. 

64. Rouse, H., Engineering Hydraulics , First Edition, John 
Wiley & Sons, Publisher, New York, 19 54. 

55. Jonsson, I. B., "Wave Boundary Layers and Friction 

Factors , " Proceedings of the 10th Conference on Coastal 
Engineering , ASCE, Vol. 1, Tokyo, Japan, 1966. 

66. Tunstall, E. B., and Inman, D. E., "Vortex Generation 
by Oscillatory Flow over Rippled Surfaces," Journal of 
Geophysics Research , Vol. 2858, 1975, pp. 6673-5691. 

67. Iwagaki, Y., and Tsuchiya, A., "Measurement of Fric- 
tion Factor in Laboratory Oscillating Boundary Layer," 
Coastal Engineering in Japan , Vol. 9, 1965, pp. 67-79. 

68. Kamphuis, J. W. , "Friction Factor Under Oscillatory 
Waves," Journal of Waterways, Harbors, and Coastal 
Engineering , ASCE, Vol. 101, No. WW2 , May 1975, pp. 
297-309 . 

59. Carstens, M. R., Neilson, R. M. , and Altinbilek, H. D. 
"Bed Forms Generated in the Laboratory Under an Oscil- 
latory Flow: Analytical and Experimental Study," 
Coastal Engineering Research Center Technical Memo. No . 
28, Washington, D. C, 1969 . 



268 



70. Inraan, D. L. , "Wave-Generated Ripples in Nearshore 
Sand," Beach Ero.si.on Eoard Tec]mi£:al Memo. No. 100 , 
Washington, D.C., 1957. 

71. Das, M. M. , "Longshore Sediment Transport Rates: A 
Compilation of Data," Coastal Engineering Research Center 
Miscellaneous Paper No. 1-17 , Washington, D.C., 1971. 

72. Krumbien, W. C, "Shore Currents and Sand Movement on 

a Model Beach," Beach Erosion Board Technical Memo. No . 
_7, U.S. Army Corps of Engineers, Washington, D.C., 19 44. 

73. Saville, T. , "Model Study of Sand Transport Along an 
Infinitely Long Straight Beach," Transactions of the 
American Geophysical Union , Vol. 31, No. 4, 1950, 
pp. 555-56 5. 

74. Savage, R. P., "Laboratory Determination of Littoral 
Transport Rates," Journal of VJaterways , Harbors , and 
Coastal Engineering , ASCE, Vol. 88, No. WW2 , 1962, 
pp. 69-92. 

75. Ingle, J. 0., Jr., The Movement of Beach Sand , First 
Edition, Elsevier Publishing Company, Amsterdam, 1966. 

76. Galvin, C. J., "Breaker Type Classification on Three 
-Laboratory Beaches," Journal of Geophysics R esearch , 

Vol. 73, No. 12, 1968, pp. 3651-3659. 

77. Battjes, J. A., "Surf Similarity," Proceedings of the 
14th Conference on Coastal Engineerxng, ASCE, Vol. 2, 
Copenhagen, 1974, ,pp . 466-480. 

78. Shay, E. A., and Johnson, J. W. , "Model Studies on the 
Movement of Sand Transported by Wave Action Along a 
Straight Beach," Institute of Engineering Research, 
University of California, Issue 7, Series 14, 19 51, 
(unpublished) . 

79. Sauvage, de St., M. , and Vincent J., "Transport 
Littoral, Formation des Fliches et Tombolos," Proceedings 

. of the 5th Conference on Coastal Engineering , ASCE, 
Grenoble, France, 1954, pp. 556-573. 

80. Fairchild, J. C, "Longshore Transport of Suspended Sedi- 
ment," Proceedings of the 13th Conference on Coastal 
Engineering , ASCE, Vol. 2, Vancouver, B.C., 1972 , pp. 
1287-1305 . 

81. Price, W. A., Tomlinson, K. W., and Willis, 0. H. , "Pre- 
dicting Changes in the Plan Shape of Beaches," Proceed - 
ings of the 13th Conference on Coastal Engineering, ASCE 
Vol. 2, Washington, D.C., 1972, pp. 1321-1329, 



269 



82. Barcelo, J. P., "Experimental Study of the Hydraulic 
Behavior of Groyne Systems," Memoria No. 35 0 , Labotorio 
Nacional de Engenharia Civil, Lisboa, Portugal , 1969. 

83. Bijker, E. W. , "Littoral Drift as Function of Waves 
and Currents," Delft Hydraulics Laboratory, Publication 
No . : 5 8 , Delft, Netherlands, 19 70 . 

84. Beach Erosion Board, Interim Report , U.S. Army Corps of 
Engineers, Washington, D.C., 19 33. 

85. Zenkovich, V. P., "Fluorescent Substances as Tracers 
for Studying the Movement of Sand on the Sea Bed; 
Experiments Conducted in the U.S.S.R.," Dock and 
Harbour Authority , Vol. 40, 1960, pp. 280-283. 

86. Bruun, P., and Battjes, J. A., "Tidal 'inlets and Lit- 
toral Drift," 10th International Association for Hy - 
draulic Research Congress , Vol. 1, London, 196 3, pp. 
123-130 . 

87. Watts, G. M. , "A Study of Sand Movement at South Lake 
Worth Inlet Florida," Beach Erosion Board Technical 
Memo . No ■ 4 2 , Washington, D.C., 1953. 

88. Caldwell, J. M. , "Wave Action and Sand Movement Near 
Anaheim Bay, California," Beach Erosion Technical Memo . 
No . 6 8 , Washington, D.C., 1956. 

89. Bruno, R, 0., and Gable, C. G., "Longshore Transport 
at a Total Littoral Barrier , " Proceedings of the 16th 
Conference on Coastal Engineering , ASCE, Vol. 1, 
Hamburg, Germany ,' Aug . 1978, pp. 128-145. 

90. Moore, G. W., and Cole, J. Y., "Coastal Processes in the 
Vicinity of Cape Thompson, Alaska, Geologic Investiga- 
tions in Support:-'Of Project Chariot in the Vicinity of 
Cape Thompson; Northwestern Alaska," Preliminary 
Report, U.S.G.S. Trace Elements Investigation Report 

753 , Washington, D.C., 1960. 

91. Longuet-Higgins , M. S., and Stewart, R.W., "Radiation 
Stress in Water Waves, a Physical Discussion with Ap- 
plication," De ep-Sea Research, Vol. 11, No. 4, 1964, 
pp. 529-563. 



92. Longuet-Higgins, M. S., and Stewart, R.W., "A Note on 
Wave Set-Up," Journal of Marine Research , Vol. 21, 1963, 
pp. 44-100. 

93. Le Mehaute", B., and Webb, L.M., "Periodic Gravity Waves 
Over a Gentle Slope at a Third Order of Approximation," 
Proceedings of the 9th Conference on Coastal Engineering , 
ASCE, 1964, -pp. 23-40. 



270 



94. Phillips, 0. M. 5 The Dynamics of the Upper Ocean , 
First Edition, Unav.epSui±y Press., .Cajnbrxdge, Mass., 
1956, pp. 22-70. 

95. Whitham, G. B., "Mass Momentum and Energy Flux in 
Water Waves," Journal of Fluid Mechanics , Vol. 12, 
No. 2, 1962, pp. 135-147. 

96. Kinsman, B., Wind Waves , First Edition, Prentice Hall, 
Englewood Cliffs, N.J., 196 5. 

97. Bowen, A. J., Inman, D. L. , and Simmons, V. P., "Wave 
Set-Down and Set-Up," Journal of Geophysics Research , 
Vol. 73, No. 8, 1968, pp. 2559-2577. 

98. Iv-erson, -H . W., "Wav-es --arid- Bi?eakers In Shoaling Water," 
Proceedings of the 3rd Conference on Coastal Engi- 
neering , ASCE, 1953, pp. 1-12. 

99. Larras, J., "Experimental Research on the Breaking 
of Waves," Ann. Fonts Chaussees , Vol. 12 2, Grenoble, 
France, 19 52, pp. 5 2 5-542. 

100. Ippen, A. T., and Kulin, G., "Shoaling and Breaking 
Characteristics of the Solitary Wave," Massachusetts 
Institute of Technology Hydrodynamics Laboratory 
Report 15 , Cambridge, Mass., 195 5. 

101. Eagleson, P. S., "Properties of Shaaling Waves by 
Theory and Experiment , " Transactions of the American 
Geophysical Union , Vol. 37, 1956, pp. 565-572. 

102. McCowan, J., "On the Highest Wave of Permanent Type," 
Phil. Mag . , Vol. 39, 1894, pp. 351-359. 

10 3. Davies, T. V., "Symmetrical, Finite, Amplitude Gravity 
Waves," Gravity Waves , NBS Circular 521, Chapter 9, 
1952, pp. 55-60. 

104. Long, R. R. , "Solitary Waves in One- and Two-Fluid 
Systems," Tellus , Vol. 8, 1956, pp. 460-471. 

10 5. Laitone, E. V., "Limiting Conditions for Cnoidal and 
Stokes Waves," Journal of Geophysics Research , Vol. 
67, No. 4, April 1962, pp. 1555-1564. 

106. Dean, R. G., "Breaking Wave Criteria: A Study Employing 
a Numerical Wave Theory," Proceedings of the 11th 
Conference on Coastal Engineering , ASCE, Vol. 2, 
Washington, D.C., 1968, pp. 1053-1078. 

107. Stoker, J. J., Water Waves , First Edition, Interscience ■ 
Publishers , New York, 196 8. 



271 



10 8. Munk, W. H. , "The Solitary Wave Theory and Its Applica- 
tion to Surf Problems , " ATinual of th-e New York Academic 
Science , Vol. 51, No. 3, 1949, pp. 376-424. 

109. Thornton, E. B., "Review of Longshore Currents and 
Theories , " Proceedings of the National Sediment Trans - 
port Study Conference , Newark, Del., Oct. 1976, pp - 
21-33 . 

110. Einstein, H. A,, "Bed Load Function for Sediment Trans- 
portation in Open Channel Flows," U.S.D.A., S.C.S., 
Technical Bulletin 1026 , Washington, D.C., 1950. 

111. Personal Conversatio n with R. G. Dean, J. Basillie 
and R. Bruno. 

112. Watts, G. M., "Development and Field Tests of a 
Sampler for Suspended Sediment in Wave Action," Beach 
Erosion Board Technical Memo. No. 34 , U.S. Army Corps 
of Engineers 5 Washington, D . C . , 19 5 3 . 

113. Fukushima, H., and Mizoguchi, Y., "Field Investigation 
of Suspended Littoral Drift," Coastal Engineering in 
Japan , Vol. 1, 1958, pp. 131-134. 

114. Fairchild, J. C. , "Longshore Transport of Suspended 
Sediment," Proceedings of the 13th Conference on 
Coastal Engineering, ASCE, Vol. 2, Vancouver, B.C., 
1972, pp. 1279-1305. 

115. Hom-ma, M. , and Horikawa, K., "Suspended Sediment Due 
to Wave Action," Proceedings of the 8th Conference on 
Coastal Engineering , ASCE, Mexico City, 1962, pp. 753-771. 

116. Shinohara, K., et al., "Sand Transport Along a Model 
Sandy Beach by Wave Action," Coastal Engineering in 
Japan , Vol. 1, 1958, pp. 211-223. 

117. Hattori, M., "A Further Investigation of the Distribu- 
tion of Suspended Sediment Due to Standing Waves," 
Coastal Engineering in Japan , Vol. 14, 19 71, pp. 3 2-56". 

118. Kennedy, J. F. , and Locher, F. A., "Sediment Suspension 
by Water Waves," Institute of Hydraulic Research Report 

' 15_, University of Iowa, Iowa City, 19 75 . 

119. Fairchild, J. C, "Suspended Sediment in the Littoral 
Zone at Ventnor, New Jersey, and Nags Head, North 
Carolina," Coastal Engineering Research Center Tech - 
nical Paper TI-5 , Washington, D.C., 1977. 



272 



120. Kana, T. W. , "Suspended Pediment Transport at Price 
Inlet, S.C.," Coastal Sediments '77 , ASCE, Charlestown, 
South Carolina, Nov. 1977, pp. 798-821. 

121. Bagnold, R. A., "Experiments on a Gravity-Free Dis- 
persion of Large Solid Spheres in a Newtonian Fluid 
Under Shear," Royal Society (London) Proceedings , 
Vol. 225, 1954, pp. 49-51. 

122. U.S. Naval Weather Service Command, "Summary of Synoptic 
Meterological Observations (SSMO) for North American 
Coastal Marine Areas," Vols. 4 and 5, Ashville, N.C., 

19 72 . 

123. Le Mehaute, B., and Koh, R.C.Y., "On the Breaking of 
Waves Arriving at an Angle to the Shore," Journal of 
Hydraulic Research , Vol. 5, No. 1, 1967, pp. 67-88. 

124. Walton, T. L. , Jr., and Dean, R. G., "The Use of Outer 
Bars of Inlets as Sources of Beach Nourishment Material," 
Shore and Beach Magazine, Vol. 44, No. 2, July 1976, 

pp. 13-19. 

125. Pelnard-Considere , R. , "Essai de Theorie de I'Evolution 
des Formes de Rivate en Plages de Sable et de Galets , " 
4th Journees de 1 ' Hydraulique , Les Energies de la Mar, 
Question III, Rapport No. 1, 1956. 

126. Yasso, W. E., "Plain Geometry of Headland-Bay Beaches," 
Journal of Geology , Vol. 73,1965, pp. 702-714. 

12 7. Silvester, R. , "Growth of Cenulate Shaped Bays to 
Equilibrium," Journal of Waterways, Harbors, and 
Coastal Engineering , ASCE, Vol. 96, No. WW2 , 1970, 
pp. 275-287. 

12 8. Johnson, D. W. , Shore Processes and Shoreline Develop - 
ment , First Edition, Hafner Publishing Company, New 
York, 19 72. 

129. Mehta, A. J., and Christensen, B. A., "Incipient 
Motion of Shells as Dredge Material Under Turbulent 
Flows , " Seminar XVII Congress of International Associa - 
tion for Hydraulic Research, Baden-Baden, Germany, 
Aug. 197 7. 

130. Wiegal, R. L., Oceano^raphic Engineering , First Edition, 
Prentice-Hall Publishing Co., New York, 1960. 

131. Carslaw, H. S., and Jaeger, J. C, Conduction of Heat 

in Solids, First Edition, Clarendon Press, Oxford, 19 59. 



273 



132. Abramowitz, M. , and Stegun, I. A., Handbook of 
Mathematical Functions , First Edition, Dover Publica- 
tions, Inc., New York, 19 72. 

133. Dean-,- R. G., and Walton, T. L. , "Chapter I, Geology of 
Estuaries," Proceedings of the Second International 
Estuarine Research Conference , Academic Press, 1973. 

134. O'Brien, M. P., "Estuary Tidal Prisms Related to 
Entrance Areas," Civil Engineering, Vol. 1, No. 8, 
1931, pp. 738-739. 

135. O'Brien, M. P., "Equilibrium Flow Areas of Inlets 

on Sandy Coasts," Journal of Waterways, Harbors, and 
Coastal Engineering, ASCE, Vol. 40 No. WWl , 196 9, 
pp. 823-839. 

135. Johnson, J. W. , "Tidal Inlets on the California, 

Oregon, and Washington Coast," University of California, 
Hydraulic Engineering Laboratory Report HEL 24-12 , 
Berkeley, California, 19 72. 

137. Jarrett, J. T. , "Tidal Prism-Inlet Area Relationships," 
U.S. Army Waterways Experiment Station, Vicksburg, 
Mississippi, 1974 (unpublished). 

138. Bruun, P., and Gerritsen, F., Stability of Coastal 
Inlets, Vol. 1 and 2, North Holland Publishing 
Company, Amsterdam:, 19 6 6 

139 . Sea Grant Nearshore Sediment Transport Study Workshop 
on Instrumentation for Nearshore Processes , Lajolla, 
California, June 19 77. 



APPENDICES 



APPENDIX A 

DERIVATION OF LONGSHORE ENERGY FLUX PER UNIT LENGTH SHORELINE 



The derivation for P,^ proceeds as follows: Assume a 
coast with contours that are parallel to a straight shore- 
line (Figure A-1) . Waves approaching this coast are assumed 
to -bfi. -described by linear..- .smaJJ. amplitude theory. In gen- 
eral, a wave crest that makes an angle with the shoreline 
when in deepwater will refract to make an angle a at some 
shallower depth (Figure A-1) , where a is related to by 
Snell's law. In what follows, the subscript, o, refers to 
deepwater conditions. 

The path of a wave passing through point i is shown on 
Figure A-1 as the dashed orthogonal labeled "wave path." 
The flux of energy between orthogonals in the direction of 
wave travel at point i is given by 
P. = C 

1 ^ ^ (A-1) 

= (n CE)^£^, 

where C is the wave group velocity, C is the wave phase 
S 

velocity, n = C /C, E" is the energy density, the total 
average energy per unit area of sea surface, 

r = CA-2) 



275 



Orthogonal 2 



Wave Crest 



R3int 



■ Wave Ray 



u Bottom 
Contours 




Beach 



Figure A-1. Definitions- for Conservation of 
Energy Flux for Shoaling Wave 



277 



where y is the weight density of water (64.0 lbs/ft for 
sea water) and H is the wave height. (This wave height is 
the height of a uniform periodic wave). 

Small amplitude theory is assumed, energy is assumed 
constant between orthogonals in this approach and does not 
spread laterally across wave orthogonals. Therefore, the 
energy flux in the direction of wave travel must remain 
coiistant.-. beiween . ,Qrthogoxia2^-,.,-±ha.t . j-s, betweezL deepwater and 
point i, 

P = P. = constant, (A-3) 
o 1 

where is defined by equation A-1. However, the longshore 
component of P^, designated P,^, where 

P^ = P^ sin a (A-14) 

does change along the wave path, since a changes due to 
refraction while P^ is constant by equation A-3. The sub- 
script i in this appendix indicates any point on the wave 
path, including deepwater, where small amplitude theory 
holds . 

From the geometry of Figure A-1, it is obvious that 
changes with position on the wave path. However, the dis- 
tance between adjacent orthogonals, b, measured parallel to 
the coast does not change. Therefore, at any point, i, in 
the wave path 

Jl. = b cos a. (A-5) 



278 



From equations A-4 and A-5, this longshore component of 
energy flux can be written 



P„ = (E C b cos a) sin a 



(A- 6) 



For the straight parallel contours assumed, the distance b 
is arbitrary. 

Arbitrarily setting b = 1 foot (or a unit length in the 
system of measurement used), one finds that the longshore 
power per unit length of shoreline is : 



= E C sin a cos a 



or using a trigonometric identity 

P„ = E C sin2a 
^ 2" g 



(A-7) 



(A- 8) 



At breaking, where the longshore wave energy dissipated 
shoreward of breaking has been assumed to be the available 
energy for transporting sand in the longshore direction 



(ttC ), sin 2a, 
2 g b' b 



or 



£b 

at breaking 

^Ih = -16- ^gb ^% 



(A- 9) 



(A-10) 



APPENDIX B 



LONGSHORE SAND TRANSPORT MODEL DATA 
SUSPENDED SAND CONCENTRATION DATA 



280 



PL. 

X 



(X, 
X 



a 
w 
o 
o 

o 



X5 


CN 






• 




CM 


■e- 


CD 


c 


c 


CO 


•H 


« 


O 


CO 



















4-( 



+-> 

CO 

<u 

H 



o 



(£> rH 

(j> CO r- 



CO 



CD 00 

LO CD LO lO 



CO CD CD 

J- CD r- 



CO H iH 



CD (D CO 
CNHrHr--<NJ-:d-CM 

oooooooo 



H H H H 



CD <N 

C7) CO CN 

CO CN CN CN 

o o o o 



CD 

o CD cn CD 
Ln CO CO o 
o o o o 



CD CO 

CNCNcotr)Lnj-cnco 

HHHiH^r-IHrH 



=i- ^ ^ 

CD CO CD CD 
rH rH H H 



CO iH rH 
CSI CN CN H 



oooooooo 



O O O O 



O O O O 



oj CO r~ cn cn 
J- zl- CD Ln LO CD LO 
O O O O O O O 



r-~ CO CN CN 

Ln zt- Lo in 
o o o o 



^ r-- cn CD 

C-^ CD CD 

o o o o 



r-i<-i<-\r-ir-ir-in-{r-\ 



H rH H H 



H H H H 



cn lO Ln CO CN H 

HCNCNCNCNCNCMCM 

oooooooo 



CN CN CN CN 

o o o o 



CN CN CN CN 
O O O O 



CD r-l CN iH CO CD Ln 
HHrHinj-cDCvlCO 
OOOCMCNHCNCN 



CO Ln Ln 

CN CO CO 

CN rH H 



CD 



00 Ln 
.H H O 



^ Ln CO 

rH cn 

cD LO Ln 00 

o o o H 

o o o o 



CO CN rH (D 

O O cn 

H CO CN 04 

O O O O 



CD CD H C~~ 

CD O CO 

Jd- CO CO 

O O O O 



00 t> H 

00 Ln C71 cvi 

CN C4 CN H 

O O O O 



C7) 

o 



CO 



Cvl iH 



CO 

O H 
O O 



(J) 



o o 



^ CD 

o o 



CN OJ 

o o 



C^J 00 

cn Ln 
o o 



CO H 

Ln Ln 

H H 

o o 





J- 


CO 


cn 


Ln 


o 


O 


Ln 


in 


Ln 








cn 


Ln 


LO 


CD 




CN 


CO 


CO 


H 


o 


CO 


CO 


O 


cn 




CO 


CO 


O 


cn 


CD 


LO 


Ln 


O 


H 


H 


H 


CN 


CN 


H 


CN 


CN 




H 


r-t 




CO 


CN 


CN 


H 


--I 


CN 



Ln 


CN 


iH 


CO 


in 


00 


H 




00 




CN 


Ln 




rH 


Ln 


Lrt 




CD 


c- 


CD 


CD 


CO 


CO 


CD 


r~ 


CO 


00 


CO 


Ln 


Ln 


Ln 


Ln 


Ln 


rH 


H 


O 


o 


o 


O 






H 




-H 


rH 


r-i 




rH 


.H 


H 


rH 


rH 




rH 


Ln 






CD 


CO 


CD 


CD 


ijO 






Ln 


LO 










LO 


lO 




co 


CO 






ca 


ca 


CD 






o 


o 








rH 


CN 


CN 


rH 






































H 


H 












■H 


-H 




H 


rH 


rH 


rH 


rH 


iH 


rH 


cn 


CO 














O 




CN 


CO 








O 


CD 


r~ 


H 


CO 


CO 


H 


CN 






CO 


H 




rH 


rH 


c-~ 


CD 


cn 


rH 


rH 


rH 


1 

o 


o 


1 

O 


1 

O 


1 

O 


1 

o 


1 

o 


1 

o 


1 

O 


1 

O 


O 


1 

o 


! 

O 


I 

O 


1 

O 


1 

O 


1 

O 


1 

O 


CO 


CO 


CO 










J- 


J- 








Ln 


Ln 


Ln 


Ln 


Ln 


Ln 



o 

CO 



o 
II 



o 

cu 

CO 



4h 

H 
O 
II 



o 

Ln 



281 







00 


CD 


(J) 






CN 








CO 










l-H 


Ph 


o 


r— i 


iH 




to 


CO 


CN 


CO 












zt 




X 


• 


* 


• 


* 




* 
























CO 


CO 




CO 




CO 






CN 








CD 












cn 


CO 




CO 


CO 


















CX3 


rH 


o 




CO 


CN 


o 






lo 








CO 




CO 


o 


CO 








CD 


CO 














X 




















CN 




CO 




H 






H 


to 


H 






rH 








CO 














CN 










H 




CN 








LO 




CO 


X 






CO 


CO 




O 




1 

r-1 




CO 




00 












CN 


CN 


rH 




CN 


*H 


(H 




CO 




1 




r^ 












* 


■ 


' 


* 


* 




' 










Xi 
































8 






















CO 




O 






tn 






















CN 




CO 




CN 


o 


















o 




r-1 










o 




































o 


o 


O 


O 


o 


O 


O 
















o 
































w 
































0) 
































Xi 
































a 


• 
































-e- 
















CO 




00 








CN 


c 


C 


CO 


CO 










CN 


<£> 




H 




Oi 




CD 


•H 




CN 


CO 


■d" 


CN 


00 


CO 


CN 


o 








CN 




1 

1 1 


M 


■P 


O 


o 


o 


O 


o 


o 


O 


o 




O 




O 




o 


































(1) 
































ca 


CO 


CO 


CO 


CO 


CO 




CO 


^ 














C 


CO 


CO 


CO 


CO 


CO 


CO 












CO 




CO 




H 


rH 


H 


rH 


rH 


H 


H 


o 




O 




o 




o 


■P 














• 










































CO 




o 






CO 


H 


CO 


CO 


ct" 


CO 


CO 






CO 




CO 








S 




<H 




rH 


»H 


1 — i 


iH 


iH 




o 










Mh 


o 


O 


o 


o 


o 


o 


o 


o 




o 














































<^ 






CD 


fH 




LO 


CO 
















=< 








LO 






o 


CO 




CO 




LO 








* 


• 


* 


" 


• 


• 


* 






• 
















CO 


CO 


CO 




CO 




CO 




LO 




o 




























00 




CN 














uo 


CN 


CO 


H 


CTi 










CO 










r— 1 


o 




CN 


CN 


CO 




CO 








LO 




LO 






" 






• 






* 
















1- 


H 


o 








CO 




1 — i 


CN 




CO 








CO 






H 


H 




















H 










O 


o 


o 


O 


0 


0 


O 


0 




0 




0 




O 












CD 




in 




CO 




cn 




tD 




CO 






































o 


CO 


CD 


CO 


CN 


rH 


CO 


H 




CD 




t> 




LO 


s 






rH 




H 


H 


















a: 








































CO 




CO 




LO 




CO 














CN 


J" 


o 


CO 




LO 






LO 








iH 






• 


■ 


* 




■ 


• 




• 




* 




* 




* 






CN 


CO 


CO 


CO 


CN 


CN 


CO 


CO 








CO 


























A 








n 












CO 










cn 


CD 




cn 


rH 


CO 


LO 


4-" 
















CD 


CD 




cn 


o 


o 


O 


W 


UD 


H 






CO 


zT 


LO 


iH 


1— i 


o 


1 — ! 


CN 


CN 


CN 


(U 


CO 




CO 






CD 


o 






It 


rt 






A 


E-i 






o 








rH 


CN 


03 


rH 


CO 


o 


CN 


zt 




















CO 


CO 




CTi 


o 


O 


O 




















iH 


iH 




i—i 


CN 


CN 


CN 










CO 








CO 


CO 










CO 








< 


< 


< 


< 


< 


< 


< 


< 


< 




< 




< 








Q 






Q 


Q 








Q 




Q 




a 








CQ 






pq 


cn 


pq 


cq 


pa 


CQ 








m 
















:s 






CO 






to 




CO 








w 




w 


W 


w 


w 




CO 






CO 




CO 





282 



It) 
a 

H 
•H 

o 

b4 



CO 
I 

m 
m 

r-i 

I 



X 



a 
to 
o 
o 

X 

o 













a 


« 








CM 


-e- 








c 


C 






OD 


•H 






=t 


CO 


CO 




O 


O 


O 












0) 











CO. 

C 

to 



CO 

<u 

E-" 



C- H O) 

cs H 



ID 

CO CO iH 
(N H O 



ID 
CM CD 
CN H O 



O O O 



Ln 

H .H O 



CO CO CD 
rH H H 
O O O 



CN 

r~ CM 
O CD CN 



CXI H 

^ m m 

H H O 



O O 

CD 



O 
CN 



CO CD 
CD CN ,H 
CO CO CN 



LO CD 
OJ CO CO 



.H H H 



CD O CO CD 

CD CD U1 UO 

• • I I 

CO r- iH oj 



CD CN 
CD ^ rH 



CN^ rH CD in 



H CD o in 
J- CO Jd- H 



CN in (J) r- 

C\l CN H H 



o o o o 



cn i> 

CO CO CN CO 

o o o o 



H H H 



CD CD H 
iH O O rH 

o o o o 



r~ LO CO rH 

CXI en iH cn 



H CO CN 



CD 

CD en CD CD 

o Ln CO i> 

r^ H CN 



0 0 0 0 

CO CD O 
H H H 



iH ^ O ,H 
LO CD CD J 



CO CO CD 
H .H O rH 



CN Cv| CO CN 



o o o o 

CD CD CD CD 

till 

CM ^ Ln CD 



o 
a; 

CO 

^^ 

o 
II 



4-> 
4h 

CD 

o 
o 
o 



o 

LO 



CO 

o 

CM 



283 



X 



X 



J- 












CJ) 


H 


r- 


CO 


CO 


CO 


LO 


H 




CN 




CN 


CO 


CO 


Ln 


CO 


CN 


04 


CN 


CM 


cn 


J- 


LD 




CM 


in 


00 


00 


Ln 


CO 


CM 




CO 


H 


H 


O 







J- 


CO 




o 


cn 


co 


CO 


CO 


m 


00 


o 




CN 


CO 


CM 


CO 


CO 




CN 



a 

CO 

o 
o 

Xi 

a 



CO 
H 

o 
o 



o 



o 

i 



o 
o 



o 
o 
o 



o 
o 
o 



J- 

o 
o 



c 

•H 

cn 



-©- 
c 



o 









o 


CO 


CN 




in 


CD 


r- 






o 


O 


O 


o 


o 


O 



cn 
c 



tn 

0) 







r-- 










CO 


CD 


CD 


CD 


CD 


CD 


CO 


o 


O 


o 


O 


O 


O 


O 


CO 


CD 


CO 


CO 


CO 


CD 


CD 


o 


O 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




in 






ID 


cn 


en 


.H 


O 


CO 


CM 


CT> 


CO 




IT) 


CD 


H 


o 




:t 


CM 


H 






rH 








CO 




CO 


CO 






00 


















CD 


CD 


CO 




CO 


CM 


-H 














o 


0 


0 


o 


o 


0 


O 


r-l 


LO 


O 


H 


CO 


CD 




CN 




CM 


CO 

1 


CO 

1 


1 


H 


1 

I> 


1 

CO 


1 

CM 




o 










H 


H 




CM 


























CO 






CM 


-H 


CM 


■H 


1 


-H 
1 


rH 
1 


H 


1 


1 


1 

CD 


CO 


LO 


un 
















CM 


o 




O 








H 






H 


CD 


t-^ 
1 


CD 
1 


1 

cn 


1 




1 

LO 


I 

CD 




in 


CD 


CO 




CJ> 




CO 


J- 




-H 


CD 


rH 
1 




H 
1 


1 


1 

r- 


CO 


1 

CD 


CD 


CN 


CM 


CO 












r-l 




?j 




u 




>, 


>, 


C 




n3 


a. 




rd 


m 


3 




s: 


< 


< 






1-^ 



1 

o 

CO 



i 



o 
II 



a 

0) 
CO 

■H 



CM 
O 
II 



o 
in 



281+ 



Table B-5. Kana Data. 



SERIES STA H(ft.) T(sec. ) D(ft. ) Y(ft.) C(ppt) 



10107 B02 1.15 10 1.81 .33 .141 

.99 .039 

1.98 .016 

10229 C+l 1.4-8 9.5 2.38 .33 .306 

.99 .097 

1.98 .038 

10343 PIl 1.48 8.5 2.70 .33 .323 

.99 .175 

1.98 .083 

10564 PI9 1.80 12.0 2.86 .33 .132 

.99 .081 

1.98 .028 

10668 CAl 1.96 11.0 3.11 .33 .259 

.99 .192 

1.98 .089 

20109 BIl 1.96 8.0 2,94 .33 1.241 

.99 .789 

1.98 .351 

20547 CAl 3.61 9.5 5.08 .33 1.244 

.99 .702 

1.98 .647 

3.28 .361 

20658 BU2 2.95 8.5 4.77 .33 0.499 

.99 0.278 

1.98 0.194 

3.28 0.113 

20767 PI9 2,30 7.0 3.61 .33 .258 

.99 .137 

1.98 .059 

30216 PIl 1,80 11.0 2.70 .33 .595 

.99 .301 

1,98 .218 

30218 PIl 1.80 11.0 2.86 .33 .376 

.99 .221 

1.98 .081 

30442 BU2 1.97 11.5 5.20 .33 .707 

.99 .280 

1,98 .073 

40214 PI9 1.80 9.5 4.03 .33 .149 

.99 .064 

1.98 .033 



285 



Table B-5. Fairchild Data. 



STA. H(ft.) T(sec) D(ft.) Y(ft. ) C(ppt) 

234-N 2.18 6.6 4-. 70 0.20 0.328 

2.18 6.6 ^.72 0,20 0.461 

1.82 9.2 4-. 74 0.39 0.4-52 

1.82 -9.2 4.76 0.39 0.457 

2.00 8.0 4.78 0.39 0.292 

2.00 8.0 4.77 0.39 0.514 

1.73 6.9 4.75 0.59 0.183 

1.73 6.9 4.74 0.59 0.777 

1.73 ..6,-9 4.72 0.78 0.105 

4.72 0.78 0.320 

2.00 8.7 4.61 0.98 0.463 

2.00 8.7 4.59 0.98 0.185 

2.00 8.7 4.52 1.18 0.235 

2.00 8.7 4.48 1.18 0.351 

1.92 7.2 4.30 . 1.37 0.253 

1.92 7.2 4.28 0.20 1.774 

1.71 6.8 4.11 0.20 1.438 
2.12 6,7 3.88 0.39 1.589 
2.12 6.7 3.87 0.39 0.951 

325-S 2.40 4.4 3.48 0.19 1.353 

2.40 4,4 3.49 0.19 1.429 

2.72 6.7 3.52 0.39 1.036 
2.71 6.7 3.66 0.39 0.783 
4.18. 5.0 4.15 0.58 1.552 
4.18' 5.0 4.19 0.58 1.183 
4.18 5.0 4.25 0.77 0.915 
3.54 5.3 4.31 0.77 0.915 
3.54 5.3 4.45 0.97 0.818 
3.51 5.1 4.47 0.97 0.689 

325-S 3.51 5.1 4.57 1.45 0.515 

3.51 5.1 4.59 1.45 0.736 

4.67 1.93 0.876 

4.69 1.93 0.875 

2.95 5.4 4.32 2.42 0.481 

2.95 5.4 4.83 2.42 0.409 

2.99 5.5 4.86 2.90 0.392 

2.99 5.5 4.87 2.90 0.547 

3.00 5.4 4.93 0.19 3.343 

3.00 5.4 4.94 0.19 3.228 



286 



Table B-6. Continued. 



STA. H{ft.) T(sec) D(ft.) Y(ft.) C(ppt) 

354-N 2.08 5.5 4,80 0.59 1.000 

2.08 5.5 4.78 0.59 0.891+ 

1.86 5,4. 4.59 0.78 1.125 

1,86 5.4 4.57 0.78 0.933 

1.86 5.4 4,52 0.98 0.927 

1.86 5.k 4.49 0.98 0.584 

1.65 5.5 4.23 0.20 1.502 

1.55 5.5 4.21 0.20 1.285 

1.40 5,5 4.07 0.39 1.178 

1.40 5.5 4.05 0.39 1.441 

354-N 2.20 5,8 3.91 0.59 0.681 

2.20 5.8 3.86 0.59 1.568 

1.65 5.8 3.75 0.78 0.703 

1.65 5.8 3.74 0.78 0.425 

1.65 5.8 3.62 0.20 1.101 

1.64 6,0 3.60 0.20 1.714 
1.49 9.,9 3.53 0.39 0.692 
1.49 9.9 3.51 0.39 1.613 
1,1+9 9.9 3.50 0.39 1.438 
1.72 5.0 3.47 0.39 1.200 
1.72 5.0 3.41 0.39 0.728 

326-N 1.85 9.1 2.33 0.20 0.456 

1,85 9.1 2.34 0.20 2.181 

1.85 ■ 9.1 2.43 0.39 1.454 

1.75 8.8 2.44 0.39 0.923 

1.75 8.8 2.55 0.59 0.710 

1,70 9.1 2.58 0.59 0.865 

1.65 9.3 2.66 0.78 0.765 
1.65 9.3 2,72 0.78 0.147 
1.82 9.3 2.76 0.98 0.288 
1.82 9.3 2.77 0.98 0.537 

326-N 1.90 9.2 3.51 0.59 0.368 

1.90 9.2 3.54 0.59 0.550 

1,90 9.2 3.60 0.78 0.253 

1.90 9.2 3.64 0.78 0.515 

1.77 9.3 3.73 0.98 0.158 

1.77 9.3 3.74 0.98 0.271 

1.53 10.0 3.57 0.39 1.473 

1.53 9.9 3,44 0,39 1.992 

1.45 9.2 3.34 0.39 1.548 

1.45 9.2 3.33 0.39 0.993 



287 



Table B-5. Continued. 



STA. H(ft.) T(sec) D(ft.) Y{ft.) C(ppt) 

326-N 



1.68 


9.9 


3.87 


1.47 


0.145 


1.68 


9.9 


3.87 


■ 1.47 


0.148 


1.59 


8.8 


3.91 


1.96 


0.033 


1,59 


9.8 


3.92 


1.96 


0.294 


1.57 


9.3 


3.94 


0.39 


1.206 


1.57 


9.3 


3.93 


0.39 


0.951 


1.57 


9.3 


3.90 


0.39 


0.588 


1.57 


9.3 


3.89 


0.39 


0.593 



3i+0-N 0.96 10.0 2.76 0.19 0.180 

0.96 10.0 2.75 0.19 0.115 

1.09 10.1 2.77 0.39 0,081 

1.09 10.1 2,77 0.39 0.013 

1.09 10.1 2.77 0.58 0.052 

1.01 9.6 2.77 0.58 0,113 

1.01 . 9.6 2.79 0.77 

1.01 9.6 ■ 2.84 0.77 0.031 

1.02 9.6 ' 2.85 0.97 0.085 
1.02 9.5 2.87 0.97 



340-N 1.02 9.6 2.94 1.45 

0.81 9.8 2.94 1.45 

0.81 9.8 3.03 1.93 

0.82 9.4 3.05 1.93 

0.82 9.4 3.13 2.42 

0.82' 9.4 3.14 2.42 

325-N 



0.67 


10.1 


2.47 


0.58 


0.100 


0.67 


10.1 


2.45 


0.58 


0.091 


0.57 


10.1 


2.44 


0.78 


0.030 


0.70 


10.2 


2.43 


0.78 


0,034 


0.70 


10-2 


2.43 


0.97 


0.048 






2.43 


0.97 


0.078 






2.44 


1.46 








2.45 


1.46 




0.52 


10.3 


2,47 


1.95 




0.52 


10.3 


2.49 


1.95 


0.046 


0.55 


10,0 


2.54 


0.20 


0.311 


0.55 


10.0 


2.55 


0.19 


0.347 


0.56 


10,5 


2.54 


0.39 


0.125 


0.55 


10.5 


2.65 


0.39 


0.065 


1.15 


10.1 


2.83 


0,58 


0.039 


1.16 


10.1 


2.85 


0.58 


0.030 



288 



Table B-6. Continued. 



STA. H(ft.) T(sec) . D(ft.) Y(ft.) C(ppt) 

258-N l.i+U 10.3 2.54 0.19 1.274 

1.4-4 10,3 2.55 0.19 2.380 

1.20 10.1 2.53 0.39 0.283 

1.20 10.1 2.54 0.39 0.498 

0.74 10.4 2.71 0.58 0.165 

0.74 10.4 2.71 0.58 0.158 

1.43 9.9 2.77 0.77 0.076 

1.43 9.9 2.77 0.77 0.174 

1.07 11.8 2.76 0.97 0.078 

1.07 11.8 2.76 0.97 0.112 

1.30 5.3 2.74 1.45 0.094 

1.30 5,3 2.73 1.45 0.102 

1.30 5.3 2.70 0.19 1.712 

0.99 10.1 2.69 0.19 0.751 

366-N . 4.30 1.45 0.083 

4.26 1.93 0.099 

4.26 1.93 0.047 

1.87 8.2 4.24 2.42 1.073 

1,87 8,2 4.22 2.42 0.087 

2.12 8.0 4.21 2.90 0.148 

2.12 8,0 4.21 2.90 0.097 

2.06 8.5 4.20 0.19 1.474 

2.06 8.5 4.20 0.19 1.477 

1.92 8.1 4.21 0.39 0.207 

1.92 ■ 8.1 4.22 0.39 0.015 

2,12 7.5 4.23 0.58 

2.12 7.5 4.26 0.58 0.040 



APPENDIX C 

ANALYSIS OF SSMO WAVE HEIGHT, PERIOD, AND DIRECTION RANGES 

The purpose of this appendix is to describe the manner 
in which the groupings of wave data listed in the SSMO 
volumes were handled for computations of longshore energy 
flux. 

Wave Height 

For the SSMO data, a representative value of H^ must 
be chosen for each interval of wave heights contained in 
SSMO Tables 18 and 19. Since energy is a function of wave 
height squared (in linear theory), a representative value of 
H^ for a given range of H^ values should be based on the 
mean square root value of the wave height over the range. 

Consider the probability of occurrence of a wave with 
specific height H as equal to p(H) in the range H^ to H2. 
The energy represented in this band of wave heights is 
proportional to H;^ the mean value of a representaxiive wave 
height squared where : 

J p(H) H^ dH 

h' = ^ 

r fH2 

p(H) dH 

^-1 



289 



290 



Since pCH) is not known,, it J_s .cnnsi dered uniform, 
which is reasonable if the wave height range < H < H2 
is small. The equation then becomes: 



H^dH 



1(H2 - hJ) 
3(H2 - H^) 



(II-2) 



dH 



Taking the square root of this value, 

1/2 



H 



1(H2 - hJ) 
_3(H2 - H-^)_ 



(II-3) 



Using Equation (II-3), representative values of were 
found for the corresponding ranges of given in SSMO data, 
and are summarized in Table C-1. 



Wave Period 

Representative values of T were assumed to be the 
average of the SSMO period ranges, and are given in Table 
C-2. For T > 13.5 seconds a representative value of T = 16 
seconds was assumed. 



Wave Direction 

Directional observations as recorded in the SSMO volumes 
are given on eight points of the compass and thus correspond 
to eight M-5° sectors of the compass. In the computation of 
the longshore energy flux, the midpoint of the sectors, as 



291 



Table C-1. Representative Values of Wave Height Used in Computation of 
Longshore Energy Flux. 



Actual SSMO Range of Heights Height Used in 

SSMO Coded Height (feet) Computation (feet) 



<1 

1-2 


.82 - 


0.82 

2.46 


0.47 
1.71 


3-1+ 


2.46 


4.10 


3.31 


5-5 


4.10 


5.74 


4.94 


7 


5.74- 


7.38 


5.58 


8-9 


7.38 


9.04 


8.22 


10-11 


9.04 


10.70 


9.85 


12 


10.70 


12.30 


11.49 


13-16 


12.30 


15.50 


13.98 


17-19 


15.60 


18.90 


17.25 


20-22 


18.90 


22.15 


20.53 


23-25 


22.15 


22.43 


23.81 



Table C-2. Representative Values of Wave Period Used in Computation of 
Longshore Energy Flux. 



SSMO Coded Period Used in Computation 



Period 


Actual 


SSMO Range of Periods 


of Longshore '- 


<5 


0 


< T < 5,5 


3.0 


6-7 


5.5 


7.5 


6.5 


8-9 


7.5 


9.5 


8.5 


10-11 


9.5 


11.5 


10.5 


12-13 


11.5 


13.5 


12.5 


>13 


13.5 


CO 


16. 



292 



givexi in the SSMO data .by ..tlie.- eight pn±Qts of the compass, 
were used as the representative values of 6 for direction of 
wave approach. When a representative wave having a given 
frequency was parallel to the coastline, the corresponding 
sector of waves was divided into two parts , one being de- 
leted from the computation and the other approaching the 
coastline from the midpoint of its half sector with the 
coTTresponding frequency halv-ed- (see Figure C-26). Wave data 
in octants with midpoints in the off-shore direction 

(> e + SQ°) for a given coastline orientation have been 
n — 

deleted from the drift computations. 

In the SSMO data it was ascertained that a considerable 
number of the original observations were taken on the 3 6 
points of the compass, and, when reduced to the eight points 
of the compass in the SSMO tables, a skew of the wave direc- 
tion was introduced. This skew amounts to a ten degree 
shift clockwise, and has been accounted for in the results 
of the littoral drift computations. 



293 




Wove data offer modification 



Figure C-26. Modification of Wave Data for 
Waves Parallel to Coastline 



Average Annual Total Littoral Drift 
Diagrams Along the Florida Peninsula 



295 



I I 1 i 




^ (13 

U T3 

n •!- 

CQ O 



3 i- 

-p > 

M- -t- 
•I- oc 

Q 1/5 

I— C 

to ^ 
s- o 
o o 

+J • 
•r- +J 

—I 00 

r- O 
+J 

O -C 

1— o 

t— CD 
iB CQ 
3 

<a: -r- 

oj c: 
CD /a 

s_ s_ 

OJ 0) 
> Lj- 



O 



o 



-M 
C 

s_ 
o 



I 

1- 

C7) 



1 
1 
I 
1 



296 






«3 




TJ 




•r- 


o 


S- 




o 


a; 




CQ 




j= 




4-' 




•r- 


0) 




r— 






■4-) 


1 ! 


If- 




•r- 


O) 


i_ 


c 


c:. 






+-> 




(/I 




=3 


s_ 


cn 


o 




-M <: 


+j 




"r- 




_J 






on 






(T3 


o 


+J 


+-> 


O 




1— 


s- 








> 


to 






a: 


c 




c 


CO 


■< 






c 




C7 


o 


ra 








0! 




> 


+-1 






1 


o 






c 




o 


o 


•p- 


•r- 


4-> 


+J 


(13 


rn 


4-! 


•r- 


C 




O) 






> 


s- 




o 


CM 




1 

O 




(U 








3 





■r- 
U- 



297 








4-1 




4-1 






M- 






OJ 








si 




O 


o 




+-> 


4-1 










+-> 


4-1 


Q 






-z. 






LU 


L 


tl 


CJ3 


Q 


Q 


UJ 






_J 








> 


> 




•r— 






+J 






n3 






CD 


to 




OJ 


O 




z: 


Q. 



298 




299 



^ 1 § 



o 
z: 

CD 




a 

oj -a 

CO -i- 

s- 

jC O 
+J 1— 
■I- UU 

3 

4- ) 4J 
i+_ O) 

r— 

5- C 

a i-" 

I— c 

fO o 

S- -r- 

o +-> 
+J to 

-M ro 
•I- -Q 

_j O). 
00 

<a o 
+-> +J 
o 

1— >i 
■a 

I— <D 
<G SZ 

:3 SZ 
c <D 

<: 

OJ O- 

1. 
> 

< E 

O 

M- -r- 

O 4-> 
(O 

C +J 

o c: 

•r- a; 

+j -i- 

« s- 

•r- O 

5- 



Ln 
I 

o 



z 

=5 



1 
I 
I 



300 











+-> 






M- 






Oi 








i- 




O 


o 




+J 


+-> 










+J 






•4- 


M- 










S- 


il 




Q 


Q 


LU 






_I 








> 


> 










-l-J 


+J 




IT3 






cn 


to 




d) 


o 






a. 






<0 




-a 


u 




(T3 


s- 


O) 


o 


CQ 






Li_ 


J= 




+J 


n 




+-> 


3 


0) 






■1^ 




4- 


l-H 






U 


0) 




o 




s_ 


>— 


0) 






s- 


Q_ 


o 




-p 


-+J 


4J 


i. 


•r" 


o 


1 


u. 


, — 


o 




+J 






o 


+J 


1— 


<u 










fO 


1— ( 












o 


ct 






+-» 


OJ 




O) 




fO 




s. 






oo 


> 




< 


1 


"4- 




O 


o 




•r- 




+-> 


o 


rO 




+J 








<u 


•r- 




S- 




« 


o 


> 





(£1 



301 



LU 
C3 






113 


u 


-a 






<u 


S- 


CQ 


o 










+J 




•I — 






















S_ 




Q 








. 


o 






S. 


_J 


o 




+-> 




-u 


-M 




cn 


_1 






o 




-M 






4-> 


-P 


O 


0) 


1— 






s: 




I — I 


fO 




3 


<u 


C 


o 




S- 




0) 








a. 


CD 






+j 


s- 


s- 




o 


> 


UU 








1 


O 


c 




o 






o 


4J 


•r" 


<a 




+j 




£= 




(U 


s- 


■r- 


ITS 


S- 


> o 



302 



LU 
O 
LU 






x: 






o 






<a 






oj -a 




CQ 


i. 






o 












u. 










-P 








OJ 




•r- 






S- 


c 




Q 1— 1 


9 
a 




s- 




S- 


+J 


> 


o 


•r— 




+J 


Q. 




+J 




1* 

•8 




■-3 


o 
> 




to 


o 
2 


Tot 


+-) 

OJ 

I — 








o 




1— i 






(U 




c 


•r" 






O 




< 


rj 


1 


CD 


_i 




CD 




1 


Avera 


VI 
1 




^- 


o 




o 


•r— 

-l-J 


f 




<a 




o 


-p 






c: 


1 


at 


cu 

■r" 



303 




2: Q. \ 

\ 



SOS 




















+J 




+J 


J= 










(U 








il 




o 


o 




+-» 


-t-J 










+-> 


+J 


Q 


q- 


q- 


■Z. 


•r— 




\±1 


5- 


s_ 


CD 


Q 


Q 


LU 






_J 




(1) 




> 


> 




•r- 






+-> 


4-> 




ro 






cn 


in 




0) 


o 






a. 



305 



I ^ ^ ^ 



Q 

LU 
CD 








u 




fO 


ea 




-a 


en 


•r- 




S- 




o 




r— 


•r— 




3 








+J 




4- 


-a 


•r— 


•r— 




i. 


o 


o 




r— 


1— 


u. 


fO 




S- 


<u 


o 


CL 


4-> 




+-5 




•r- 




_] 


O 




+J 


r— 






+J 


-)-> 


<u 


o 




1— 














o 


r3 


s_ 






C 




■•-^ 








CD 


r— 






fC 




s- 






1 


> 




< 






o 






O 


4-> 




(13 


£= 


-1-J 


O 


C 


•r— 


o; 


-P 




fO 


s- 


•1— 


o 


s_ 








> 












1 

o 




<u 




u 




Zl 




CD 








U- 





306 



^ I I I I 




307 








o 




*r— 


rtJ 


+J 


■o 


(0 


•r- 


-I-' 


S- 


c 


o 




I — 






i- 




o 






O) 


J= 


o 


o 


sz 




n3 


0) 


S- 


CQ 


+-> 




SI 




UJ 


+J 




•r* 


>> 


3 






CQ 


+J 




M- 


<D 


"r- 




i. 


^ 


a 


o 




+J 


( — 


(O 


fO 


-C 


S- 


s 


o 




+-> 


+J 


-l-J 


o 


•r— 


o 


_J 


ji: 




o 






ra 


o 


•+-> 


+-> 


o 




1— 


(U 




CJ 


I — • 


sr 


res 


na 


=3 


i~ 




+-> 


£1 


s: 


<: 


LU 


C3 


>) 


cn 




(0 


an 


i. 




0) 




> 




< 


o 




o 




fO 


o 






c: 


£Z 


O) 


o 


a. 


•r- 








as 








s- 








> 












1 

o 




<u 




&. 




3 





308 




c. 
o 

•r- 

-M 10 

+J -r- 
C S- 
CU O 

•r- I — 
i- U_ 

O 

u u 
la c 

(U (t3 
CQ i- 

•4-> 

^ c: 

+-> LlI 

+J CQ 
•1- 2 

-o 

r— E 
S_ 

O • 

4- > +J 

■f— 

-J o 

<t3 <D 
-P O 

O E 
1— « 
i- 

I P 

<a c 

3 LU 

c 

<Z fO 
CQ 

■0) 

O) <D 
(0 O) 

5- J= 
<U o 
> -M 

<: ta 
2 

O 4J 

c o 

O -C 
•I- o 

+J 
« 

s- 



1 

o 



CD 



u. 



309 




c 
o 

•r- 

re 

+j re 

(U -.- 
•I- s- 

s- o 
o •— 



re oj 

CQ C 

re 

j= s- 

■1- c 

5 uj 

+-> >> 

4- re 

•I- CQ 

s- 

Q ^ 
Q. 

I— <u 
re to 
s- o 
o -t) 

-t-j • 

•I- -1-3 

_i oo 
I— o 

re ^- 
■+-> 

o OJ 

1— o 

E 

I— re 
re s_ 
rs -M 

C £= 

>> 

oj re 

CDCQ 

re 

5- 3 
<U OJ 

> i- 
< -a 
c 

O 

C -P 

o 

•r- 
+J 
<C 
•r^ 

im 

(0 



I 

3 

•r- 
U. 



310 








+-> 








Ol 








S- 


(0 


o ~o 




•r- 




S- 


u 


o 


(0 




OJ 


IL. 


CO 










C/l 




nj 




r~ 


S CQ 


■t-i 








■r- 


(/I 


i. 




a 


0) 




Q. 


r— 


rti 


ns 


o 


S- 




o 


o 


-p 




4-1 




•1— 


o; 


_J 


<j 




s= 






ta 


s- 


+-> 


+J 


o 




I— 


LU 




>> 




ro 


3 CD 










Q. 




<D 


O) 


tn 


cn 


O 


(T3 




i- 








> 


+j 


< 






1 


O 






c 




o 


o 






-P 


+J 




<a 








il 




ta 




> 





I 



311 




u 

0) -o 
CQ -t- 
i- 

J= o 

+J I— 

•I- u. 

3 

+J OJ 

>4- en 

•r~ S- 

S- O 
Q OJ 
CD 

I — 

fa • 

O CO 

•u 

•I- Q. 
_J rtJ 

o 

Its o 
O 

1— </) 

15 

C SI 

C rt3 
<Z CO 

tu OJ 
a> CL 
n3 <a 

5- O 
> I 

M- O 
O •!- 
+-> 
C n3 
O 4-> 
•r- C 
4-) O) 
(O 

•T- S_ 

J- o 



I 

0) 
S- 

:3 

CD 



312 




u 

o) "a 

CO "r- 

SI o 

-p I— 



+J CO 

<4- to 

•1- /D 

S- D_ 

S- 

I— CD 
no +J 

■4-> C 
•1- O) 



o 
o 

CO 

-p 

o 



+-> 
o 



(0 



<o <c 
s- 

> 

o 

4- -r- 

O +J 

c: 4-> 

o c 

•r- OJ 

4-1 -r- 

fS J_ 

•r- O 

S- 



CO 

I 

o 

CD 



313 




U -r- 

<u o 

CO I— 



-M •> 

•r- O) 

^- s- 

•i- 4-> 

S- c 

O LU 

r- >, 

to ra 

S- CQ 

o 

-M Q. 

•■- E 



"3 
+-> 
O 



ro 



o 

4-> 

</) 
to 
CO 



S- 

+-> 

(O to 



S- 

> 

o 

c 
o 

-M 
fO 

i. 

to 



03 
O 



c 
o 

'r— 

+-> 

to 
+-> 
c 

0) 

1- 
o 



cn 
I 

s. 



314 








4,j 




C 












s- 




o 






-o 


-E 




a 


T. 


to 


O 


<v 




CD 


[Z 






4-) 


CO 




to 


3 


(0 




Q. 


4-1 




M- 


(O 




4-> 


il 


o 








rt3 




S- 


(0 


(t3 


s- c/1 


o 




+-» 


cn 


4-5 


•r— 




CQ 


_J 






o 


r" 


+J 


«3 




+-> 


<D 


O 


O 


(— 






(T3 




S_ 


fO 


4-3 


r3 




c 


UJ 






<C 








<U CQ 


CD 


(0 


<B 


5- 


Q. 




E 


> 


(13 


< 1— 


M- 


1 


O 










O 


O 'r- 


•1— 


4-> 


+J 




fO 




•r— 




S- 




n3 




> 





o 

CM 



OJ 

S-- 



315 




u 
ta 

cQ -a 

i- 

^ o 
+J <— 

•r- U. 

5 

-M -fJ 

Z c 

O 1—1 

I— <u 

(13 O 
S_ -r- 
O C 
+j 

•+J > 



_1 o 

n3 I/) 

O «3 
I— a. 

r— id 

Z3 O 
C t/1 
C rt3 

rO 

CD 
CD 

(O CT 
S- 

oj en 
> 

<£. I 

M- £= 
O O 

SZ -M 

O fO 

■r- -t-J 

-l-J C 

fC O) 

•r- "r- 

1. S- 

<0 CD 



I 



•I— 

Ll_ 



316 



111 I ^ 




317 



•s *i 'i I 




318 




(O -r- 
CQ O 
J= U- 



3 >5 

03 

+J CQ 
•I- (/) 

%~ o 

Q " — 
S- 

1— <T3 
« o 

o c 
+-> <a 

4-5 C/1 
_1 o 
r— 

ro 4J 
+J CU 
O I— 



c: c 

C rtJ 

<a o 
s- o 

<D CQ 
> 

< 1 

^- c 
o o 

•r- 

O 13 
•r- 4-> 
+-> C 

s- s- 

IT3 O 
> 



CO 
CM 
I 

O 

<u 
%. 

Z3 



319 



? ^ B 




320 








o 




ra 




CD 




CO 








^ 






i- 


■r" 


o 


3 


LL. 


+-> 




4- 




•r- 


o 


S- 


s= 


d 






s 




o 


m 


ai 


s- 




o 


<u 


4J 


Q. 


+-> 




•I — 


o 


_J 






o 


r— 




n3 




+-) 




O 




1— 






Cl. 








(/) 


Z3 


C 




•r" 


C 


05 


•a: 


05 




•r" 


0) 3 


cn 




ta 


1 


s- 




<v 


c 


> 


o 




'r- 






4- 


ra 


o 


+-> 






c: 


<u 


o 


■r" 




s- 


+J O- 


tO 




•r" 




s- 








> 





Ln 
I 

o 

cn 



BIOGRAPHICAL SKETCH 



Todd L. Walton Jr. was born on March 17, 194 7, in 
Dayton, Ohio. He graduated from Belmont High School, Dayton, 
Ohio, and then entered the University of Cincinnati, in 
Cincinnati, Ohio. While in this cooperative college, he 
worked as an engineer with the Baltimore and Ohio . Railroad 
and the Southwestern Portland Cement Company to pay his way 
through school. He graduated in June 19 70 with a Bachelor 
of Science degree (with honors) in Civil Engineering. After 
undergraduate school he enrolled in a graduate program at 
the University of Florida and completed a Master's degree 
in coastal and oceanographic engineering (with high honors) 
in June 1972. During the years from 1970 to 1972 he also 
completed a Reserve Officers Training Course.%and was com- 
missioned as a 2nd Lieutenant in the U.S.A.R. Upon gradua- 
tion, he fulfilled his military obligation with the U.S. 
Army Engineers at Ft. Belvoir, Virginia, and joined the 
faculty at the University of Florida as an assistant 
engineer.. In the years since graduation he has taught 
classes in coastal structures and littoral processes, worked 
in research in the area of coastal hydrodynamics and beach 
erosion problems, and served as the Coastal Engineering 
Specialist to the Florida Sea Grant Program, as well as 



321 



322 



serving on a Governor's Task Force (Florida) on Beach 
Erosion and Inlet Maintenance. He then decided to pursue 
more graduate studies. 

"For thou hadst cast me into the deep, in the midst 
of the seas; and the floods compassed me about; All thy 
billows and thy waves passed over me" (Holy Bible. 
Jonah II : 3) . 



I certify that I have read this study and that in niy opinion it 
conforms to acceptable standarxis of scholarly presentation and is fully 
adequate, in scope and quality, as a dissertation for the degree of 
Doctor of Philosophy. 




B. A. Christensen, Chairman 
Professor of Civil Engineering 



I certify that I have read this study and that in my opinion it 

conforms to acceptable standards of scholarly presentation and is fully 

adequate, in scope and quality, as a dissertation for the degree of 
Doctor of Philosophy. 




T. Y. Chiu>^ 
Associate Professor of Coastal and 
Oceanographic Engineering 



I certify that I have read this study and that in my opinion it 

conforms to acceptable standards of scholarly presentation and is fully 

adequate, in scope and quality, as a dissertation for the degree of 
Doctor of Philosophy. 



J. G. Melville 

Assistant Professor of Civil Engineering 



I certify that I have read this study and that in my opinion it 
conforms to acceptable standards of scholarly presentation and is fully 
adequate, in scope and quality, as a dissertation for the degree of 
Doctor of Philosophy. 

Daniel P. Spangler 

Associate Professor of Geology 



This dissertation was submitted to the Graduate Faculty of the College of 
Engineering and to the Graduate Council, and was accepted as partial ful- 
fillment of the requirements for the degree of Doctor of Philosophy. 



June 1979 



Dean, College of Engineering 



Dean, Graduate School